Washington (CNN) President Donald Trump's frustration with the Democratic impeachment probe is boiling over , with investigators set Thursday to peel back yet another layer of what is being revealed as a broad, and possibly unlawful, behind-the-scenes scheme to pressure Ukraine for political gain.

The President's tantrum, described by top Democrats, came with his increasingly vehement denials of wrongdoing being challenged every day by testimony from current and former officials that has undercut the administration's effort to stall an impeachment process that it claims is illegal in itself.

Another day of danger looms on Thursday for the President with Gordon Sondland, the Republican donor-turned-US ambassador to the European Union, due to give a private deposition on Capitol Hill that could get to the root of Trump's backdoor dealings with Ukraine.

Sondland was a go-between linking Trump's circle to the government in Kiev, amid allegations the White House conditioned incentives -- including hundreds of dollars in military aid -- on Ukraine's willingness to open an investigation into the President's possible 2020 foe, Joe Biden.

The President's lawyer, Rudy Giuliani, is meanwhile slipping ever deeper into trouble, amid revelations about his go-it-alone Ukraine policy shop that could also come back to hurt Trump.

CNN reported Wednesday that a federal inquiry into Giuliani's activity also includes a counterintelligence probe -- to establish whether the former New York mayor's business ties to Ukraine made him vulnerable to a foreign intelligence service.

"He needs an attorney. I mean, he needs a bunch of attorneys, because Giuliani is in serious trouble," said Guy Smith, a former adviser to President Bill Clinton's impeachment defense.

"If there's a (counterintelligence) investigation going on, this is serious business. Those guys don't just look for non-paid parking tickets. This is serious stuff," Smith told CNN's Brooke Baldwin.

Trump's meltdown

Trump unleashed his fury at Pelosi during a briefing for top congressional leaders on the situation in Syria following Turkey's assault on Kurdish anti-ISIS allies abandoned by Trump

"What we witnessed on the part of the President was a meltdown, sad to say," said the California Democrat, who colleagues said the President blasted as a "third-rate politician."

The showdown followed a vote in the House that overwhelmingly condemned Trump for paving the way for the Turkish invasion, in which many Republicans broke with their standard compliance and lined up against the President.

"We were offended deeply by his treatment of the speaker of the House of Representatives, said Pelosi's No. 2, Steny Hoyer of Maryland.

White House press secretary Stephanie Grisham described the meeting differently, saying in a statement that Trump had been "measured" and "decisive" and that Pelosi "had no intention of listening."

After the meeting, the President tweeted several photos of the gathering, including one of Pelosi that he captioned, "Nervous Nancy's unhinged meltdown!" The speaker retaliated by turning the picture into her Twitter cover shot

Trump then tweeted, "Nancy Pelosi needs help fast! There is either something wrong with her 'upstairs,' or she just plain doesn't like our great Country. She had a total meltdown in the White House today. It was very sad to watch. Pray for her, she is a very sick person!"

For all the bitter political battles between Republicans and Democrats in the post-9/11 era, there has never been this level of personal animosity between a President and his enemies.

One of the chairmen leading the investigation -- Rep. Elijah Cummings, of the oversight committee -- died Thursday . It is not immediately clear how his death will impact his committee's investigation.

Cummings had been an outspoken critic of Trump, warning that his presidency posed a threat to constitutional principles.

Trump attacked Cummings in turn, calling his district a "rat and rodent infested mess."

Trump's fury at the impeachment probe had been in evidence earlier in two White House press events with Italy's President.

"People, like, that are testifying -- I don't even know who they are. I never even heard of some of them, most of them," Trump told reporters in the Oval Office.

CNN has reported that the White House is frustrated that it cannot get a handle on the impeachment drama because it learns of developments only when they seep out of closed hearings.

"I have all these people testifying. And then they leak out. They don't say the good parts, they only say the bad parts," the President said, complaining that he was not allowed lawyers in the depositions. "The Democrats are treating the Republicans very, very badly."

The case against the White House appears to have darkened this week with testimony by the White House's former top Russia official, Fiona Hill , and several career diplomats.

Hill testified that she had been advised by then-national security adviser John Bolton to inform White House lawyers of her alarm at activity by Giuliani and others, sources have told CNN.

One source said that Hill, a Trump appointee, had seen "wrongdoing" in the White House approach to Ukraine and tried to report it to officials. She was concerned that Giuliani was circumventing the State Department by seeking the removal of then-US Ambassador to Ukraine Marie Yovanovitch and pushing for Ukraine to open an investigation into the former vice president and his son Hunter Biden. There is no evidence of wrongdoing in Ukraine by either Biden.

Democratic investigators will also want to ask Sondland about new details of Hill's testimony reported in The New York Times . The paper reported that Hill had said Sondland revealed in a meeting that Trump would meet Ukrainian President Volodymyr Zelensky if his government opened the investigation he wanted.

Testimony is a new wild card for White House

Sondland's testimony before three House committees represents another wild card for the President.

A longtime Republican fundraiser, Sondland plans to show up on Capitol Hill under a subpoena despite the administration's policy of seeking to bar testimony from serving officials.

Lawmakers are eager to press him about text messages he exchanged related to Trump's July phone call with Zelensky and the freezing of foreign aid to Ukraine that cut to the heart of the Democrats' impeachment probe.

Sondland has been a player in Republican politics for a number of years. A hotelier and philanthropist, he was a late Republican convert to Trump's cause and was rewarded with his current position -- a lofty one for a non-diplomat.

He probably did not bargain for being caught in the middle of the biggest presidential scandal in decades, a factor, along with his neophyte status in Washington, that makes his testimony unpredictable.

Lawmakers may want to know, first of all, why he was involved in a close circle of policy making on Ukraine at all -- since the former Soviet state is not a member of the European Union.

State Department Deputy Assistant Secretary George Kent told the inquiry on Tuesday that Sondland, along with then-US Envoy to Ukraine Paul Volker and Energy Secretary Rick Perry, were in an informal Ukraine policy group dubbed the "three amigos," according to Democratic Rep. Gerry Connolly, a Virginia Democrat who's a member of the House Oversight Committee.

Sondland will also be questioned over a series of text messages with Bill Taylor, the top US diplomat in Ukraine, that have emerged from the impeachment investigation.

Taylor raised concerns about the US withholding nearly $400 million in US military and security aid in a text that has become a key piece of evidence in the inquiry.

The Washington Post reported that Sondland is expected to tell Congress that Trump relayed to him directly the content of a text he sent to Taylor denying any quid pro quo with Ukraine.

Democrats will seek more understanding of Sondland's conversation with the President that led up to that text message, and may suggest it is an attempt to create a record of mitigation should the texts emerge in public, as they now have.

Republicans will argue, however, that the text proves that the President had no intention of withholding military aid from Ukraine in order to force it to dish dirt on Biden.

Trump's closest allies in Congress renewed efforts to discredit the impeachment process.

"The President didn't do anything wrong," House Minority Leader Kevin McCarthy of California told reporters on Wednesday.

He also picked up Trump's attacks on House Intelligence Chairman Adam Schiff, accusing the California Democrat of lying about his contacts with an intelligence community whistleblower who first raised the alarm about the President's dealings with Ukraine.

There is "no doubt he should be censured," McCarthy said.

This story has been updated.