As chance would have it, the week before I interviewed Jeffrey Tambor I spoke to two other actors, both of whom brought up Tambor and promptly went gooey-eyed.

“Jeffrey set the tone for the whole of Arrested Development,” said Tambor’s co-star and on-screen son, Jason Bateman. “We all watched him when we filmed on the first day, and he did it totally straight. He was so bold and completely unconcerned with being likable. He taught me what you can get away with in comedy.”

A few days later, it happened again: “Just watching Jeffrey Tambor on [the 1980s cult US sitcom] The Larry Sanders Show taught me everything about comedy,” said the second actor, referring to Tambor’s glorious performance as Hank, the hard-done-by sidekick to talkshow host Larry Sanders, played by Garry Shandling. Tambor is, this actor continued, “the master”.

On a rainy Wednesday morning in London, I find the master in a hotel room, sipping on a carton of coconut water. “What more do I need?” he asks the room, and his look of almost childlike contentment suggests not much. In front of him is a table of lavishly topped bagels: “Thank u London. Honored,” he later tweets solemnly, attaching a photo of the humbling bagels.

“I have to tell you, did you know that Garry [Shandling] over the weekend tweeted ‘Watch Jeffrey Tambor in new TV show Transparent’?” Tambor asks, his round eyes getting even rounder. “Garry does not do that! I couldn’t believe it!”

Larry would not have done that for Hank, I suggest, and Tambor hoots with delight.

“No, he wouldn’t! Oh! Hank was so real to me …” Tambor is so moved at the memory that he suddenly seems on the verge of tears.

Tambor as Mort in Transparent Photograph: Beth Dubber/PR

Tambor is one of the finest American character actors working today, which explains why he is working so much – “And at the age of 70! That isn’t supposed to happen!” he exclaims, frequently. It is no coincidence that he starred in what are widely agreed to be two of the finest American sitcoms of all time – The Larry Sanders Show and Arrested Development – because his performances in them were as invaluable to their success as the writing and directing. It’s often said that the secret to Tambor’s comedy is that he plays comedy as seriously as if it was drama, and this is true. But less noted is that he takes acting in a sitcom as seriously as if he were playing Chekhov at the National. At the risk of sounding pretentious (and Tambor himself can get endearingly pretentious when talking about his acting), he completely immerses himself in the role.

“You want to feel: ‘I know that character.’ The secret of this game came late to me, and it’s that you need to work with people who you get, and I got Garry. That’s my north star,” says Tambor.

Tambor has now taken on a role for which he deserves to win, at last, the industry awards that have stubbornly eluded him. In Transparent, a series made for Amazon Prime, he plays Mort, a divorced father of three. In the first episode, he summons his children over for dinner and they assume – in classic Jewish family-style – that he’s going to tell them that he has cancer. In fact, Mort is transgender and has, for some time, been living a hidden life as Maura. “This is definitely the most transformative role of my career, and I don’t just mean in the externals,” says Tambor. “This was not something I could pull out of my Jeffrey Tambor technique bag. I kind of like not knowing how to do something – it’s more exciting.”

Sitting next to Tambor on the sofa is Jill Soloway, the writer and director of Transparent. Soloway is best known in the UK for writing and producing Six Feet Under, but she also won the dramatic directing award at Sundance last year for her first feature film, Afternoon Delight, and founded, back in the 1990s, the now legendary Real Live Brady Bunch in Chicago, in which then struggling actors including Jane Lynch and Andy Richter acted out Brady Bunch episodes. She and Tambor have the gentle ease with one another of a father and daughter, frequently referring to each other for backup. Soloway wrote Transparent with Tambor in mind: “Everyone knows how funny Jeffrey is, but he is also a great actor because he can make you feel.”

A scene from Transparent. Photograph: Beth Dubber/AP

The two of them talk at length about the transgender people who helped them during the making of Transparent. The writer and activist Jennifer Finney Boylan was an especially close guide and, because she, like Maura, transitioned relatively late in life, “a lot of who Maura became is really informed by Jenny,” says Soloway. Young twentysomething artists Zachary Drucker and Rhys Ernst acted as associate producers. “They’re part of a newer generation and they introduced us to this civil-rights movement that we’re at the beginning of,” says Soloway.

“And,” Tambor adds, “Zachery and Rhys took me on my first field trip. They took me to – what’s the name of the club?”

“The Oxwood Club,” Soloway replies, referring to a transgender nightclub in Los Angeles.

“They took me to the Oxwood. First they came to the hotel and we talked and did the metamorphosis with the wig and the makeup, and I don’t think I’ve ever been so scared in my life as I was when I walked through the hotel as a woman. It was an epiphany, because I realised, this is how Maura feels every day of her life.”

How did he feel in the Oxwood?

“Wonderful! It was revelatory. I walked in, and it was so important to see people who were just there and having fun and free and I just relaxed and danced – we danced! I got immediately that there are so many different ways of doing this [transitioning] and it was like - ” and he collapses with a happy sigh against the sofa.

Tambor was born in San Francisco, raised conservative Jewish but grew up in what he describes as “a liberal environment”. However, he was, he says, “ignorant about [transgender people] with a capital IG. So I went in with my eyes wide open. And you know, there’s so much transphobia, so much fear. I’m just happy that we can shed – ” and Tambor breaks off, overcome with emotion. At times, he seems as delicate as a just-peeled orange, full to bursting with emotions.

At the beginning of our time together, I ask Soloway what inspired her to write Transparent and she talks about how bringing understanding to transgender people is “the civil rights issue of our time, and, after Sundance, I was looking around thinking, what’s next?” But an hour later, when I ask if there was a human story that grabbed her attention she says, “Well, my own parent is trans. They came out to me as trans three years ago and pretty shortly after that, the story came to me.”

Tambor and Transparent director Jill Soloway. Photograph: Linda Nylind

Soloway wrote the show with her older sister, Faith, who is a lesbian, and the older sister on the show, Sarah, played by Amy Landecker, reignites a lesbian love affair from her youth. “There’s a lot of Faith in Sarah, and a lot of me in Ally [the younger sister],” says Soloway.

Did she talk to her parent about Maura’s emotional journey?

“Sure. My parent has their own story, and I feel so lucky they allowed me to tell this story in that it’s so public and they don’t want to be public in that way. I happen to be the person who gets to tell this story and my parent happens to be the person who inspired me with their bravery and their break for freedom.”

These days Tambor lives in New York with his wife and four young children. But he also has an older daughter from an earlier marriage, “and she’s a scholar, I care what she thinks”. One day, she came to visit her father on set and sat with him in his dressing room while he was made up as Maura.

“And she looked at me and said, ‘Dad. You’re pretty!’ And that, well … ” And once again, Tambor falls back into the couch, overcome with happiness.

• Series one of Transparent is available to watch on Amazon Prime Instant Video.