Aidy Bryant admits that she does think about the day when she’ll say goodbye to “Saturday Night Live.” “I sometimes see these new people, and I’m like, ‘I want to make room for them,’” Bryant, who has been with “SNL” since 2012, says on Thursday’s episode of the Variety and iHeart podcast “The Big Ticket.” “But I’m still having so much fun there, and I still feel challenged there, and it’s my home. It’s truly where I started. When I got hired, I had never been on camera in any form.”

While she’s racked up a slew of television and film credits since then, Bryant became a leading actor for the first time in Hulu’s “Shrill” as Annie, an aspiring writer at an alternative newspaper in Seattle who’s juggling work, romance, friends and her parents. The series, based on Lindy West’s memoir “Shrill: Notes From a Loud Woman,” has been picked up for a third season

Bryant says she always had hopes that she would play a woman who accepts her body the way it is, but she wasn’t sure it would happen. “Even when I was auditioning for ‘SNL’ I was like, ‘I wonder if they won’t hire me because I’m fat or because that’s sort of not the typical body you see on camera or there are less people for me to do impressions of, or that kind of thing,’” she says. “So then to kind of be in the position to also take on [‘Shrill’] and really spearhead it? Total dream come true.”

How much do people think that Annie is based on you?

Probably more than I’m comfortable with. [Laughs] Just the other day someone asked me about kind of a nasty sex scene we did, and they were like, “That’s based on you?” And I was like, “No, that’s not me!” But you know, there are definitely shreds of my own story in there.

Is there any real-life experience that you didn’t want to put in the show?

Not yet. I have things that I would be excited to put in going forward about being more secure with yourself. Even the experience of doing sex scenes on this show for me was a little bit of my own body journey. And I think I was maybe a little cocky going into making it where I was like, “Oh, I’m over a lot of these body issues, and I feel comfortable in my own skin, and I’ve worked really hard through therapy and friends and family to face some of these demons.” And then it was like, “Okay, now be in your underwear in front of a crew of 30 people.”

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Aidy Bryant photographed for Variety’s “The Big Ticket” podcast. Dan Doperalski for Variety

How do you balance the fat jokes to make sure people are not laughing at you, but with you?

That is hard because there are fully jokes about what she eats. But I always feel like it’s sort of just a gut-check feeling, where you’re like, “That feels good or that doesn’t feel good.” And we have other fat writers…We know the experience, and we kind of check it from within. But I also think it’s like you’re always on Annie’s side even when she’s doing bad.

Tell me about the first audition that you ever went on.

I got sent out to do an audition for a Walmart commercial, I was like 19 years old, and it was for a mother of three. I was like, “I don’t think I’m going to get this.” And then I went in there, and every other woman there was in her 40s but sort of had my body type, and I was like, “Cool. This will be my journey in this industry.”

Did you ever have an agent or manager say, “I want to work with you, but you’ve got to lose the weight”?

I didn’t have an agent tell me that, but I definitely had an agent tell me while I was in Chicago, “You’re not going to be castable for most things. You’re going to be a really specific thing.” So they didn’t tell me to lose weight, but they were like, “You’re going to be in hell for the rest of your life.”

Who else do you want to do an impersonation of on “SNL”?

Oh, my gosh, that’s a tough one. I really can’t even think of it because at this point I would say Elton John, but then I finally did him. He was my great missing piece.

If you could sit down with Sarah Huckabee Sanders, what would you ask her?

“How’s it going?” I don’t know. I would be scared to sit down with her I think.

Would you?

Well, I’ve heard tales that they didn’t love my impression over there. But yeah, I don’t think we have much in common. 

This interview has been edited and condensed. Hear it in its entirety below. You can also find “The Big Ticket” on iHeartRadio or wherever you listen to your favorite podcasts.