Washington (CNN) This week's shakeup at the Department of Justice could affect the outcome of a subpoena challenge that an associate of Trump ally Roger Stone is waging against special counsel Robert Mueller.

That quickly became clear Thursday during oral arguments at the US Court of Appeals for the District of Columbia Circuit, which is hearing the challenge from Stone associate Andrew Miller.

Judge Karen LeCraft Henderson instructed attorneys from both sides to argue their case as if the hearing were occurring 24 hours earlier, before "the events of yesterday afternoon," a veiled reference to President Donald Trump's decision to fire Attorney General Jeff Sessions on Wednesday.

Henderson, appointed to this court by President George H.W. Bush, said the three-judge panel would likely ask the lawyers to file new briefs adjusting their arguments to reflect the news.

The Justice Department shakeup affects the Russia investigation because acting Attorney General Matthew Whitaker has assumed the role of overseeing Mueller. Whitaker has previously bashed Mueller in public comments, raising concerns that he might now move to shut down or neuter the probe.

At one point during Thursday's hearing, prosecutor Michael Dreeben was asked whether Mueller's overseer could change or rescind the May 2017 order that appointed the special counsel.

Dreeben acknowledged that Mueller's boss had the power to make those changes. The line of questioning seemed to lay out a road map that Whitaker could follow to fire Mueller, though nobody explicitly stated that in the courtroom or mentioned Whitaker by name.

Outside the courthouse after the hearing, Miller's attorney Paul Kamenar more explicitly described how Mueller could be removed.

Whitaker "can either revoke the regulations altogether or amend the order ... and therefore make Mueller an at-will prosecutor ... meaning he can be fired without cause," Kamenar said. "And we think that that is the way under the Constitution it should be done for accountability."

In a discussion during the hearing about Mueller's supervision, Judge Judith Rogers noted that "what may be inappropriate to one deputy attorney general might not be inappropriate to another." The comments from the Clinton appointee were a nod to the reality that a new boss for Mueller could reach very different conclusions about whether he was running a fair investigation.

Beyond the drama unfolding outside the courthouse this week, Thursday's hearing largely focused on the substantive issue of whether Mueller had been appointed in accordance with the Constitution.

Kamenar argued that Mueller has "extraordinary prosecutorial power" and "lacks supervision" despite "acting like a US attorney at large." According to his arguments, Mueller is so powerful that instead of being appointed by Deputy Attorney General Rod Rosenstein, he is more like a US attorney -- who needed to be tapped by Trump and confirmed by the Senate.

Kamenar also made a case that Rosenstein never had the power to appoint Mueller in the first place. He said Sessions' recusal from overseeing the probe in March 2017 meant Rosenstein "could handle the investigation part but not the appointment part," which was still under Sessions' purview.

Judge Sri Srinivasan, an Obama appointee, had some sharp questions for Kamenar. They engaged in an extended back-and-forth about the case law, with Srinivasan appearing to disagree with some of Kamenar's interpretations.

For his part, Dreeben pointed to a series of court decisions and urged the appeals panel to "take heed of precedents" and fully uphold Mueller's powers. Before Thursday's hearing, four district-level federal judges had upheld Mueller's appointment and constitutional authority.

Dreeben also pushed back on the notion that Mueller lacked adequate supervision.

"He is aware of what we're doing," Dreeben said in an apparent reference to Rosenstein. "It is not the case that the special counsel is off and wandering in a free-floating environment."

Thursday's hearing came after months of legal wrangling. Mueller's team subpoenaed Miller in May, demanding that he testify before their grand jury in DC. Miller eventually beefed up his legal team and, with funding from conservative legal groups, decided to fight the subpoena.

His case was heard by Chief Judge Beryl Howell of the DC District Court, who eviscerated his legal arguments against Mueller's powers in a sweeping ruling in July. She ordered Miller to appear before the grand jury, though that hasn't happened yet, since the case was appealed.

Miller is one of about a dozen Stone associates who have testified before the grand jury, given voluntary interviews to investigators or received subpoenas from Mueller's team. Investigators have ramped up their scrutiny of Stone in recent months and have focused on his supposed contacts with WikiLeaks, his political committees that were active in 2016 and more.

Stone and Miller have known each other for years, with Miller occasionally working as Stone's driver, traveling aide and "tech guy" who handled Stone's website. A self-identified Libertarian, Miller has also been a legal marijuana farmer in California and currently paints houses in Missouri -- and he says he knows nothing about Russian collusion or election interference.

Stone, who was not at Thursday's hearing and was not a party in the proceedings, vehemently denies colluding with Russia during the 2016 campaign and questions whether the Kremlin even interfered, despite the consensus from US intelligence agencies.

This was the first public appeals hearing regarding the Mueller probe. Kamenar said after the hearing that he would "certainly" appeal to the Supreme Court if the appeals judges rule against him, though experts say it's unclear whether the high court would take the case.

"We believe at that point with this current bench that they would find that Mueller is basically acting as a rogue type of a prosecutor," Kamenar said. Asked if he was referring to the presence of newly minted Justice Brett Kavanaugh, Kamenar replied, "Absolutely."