Kate McKinnon was walking across New York City, alone, with Hillary Clinton's voice in her ears. It was the fall of 2017, and she was listening to the audiobook of Clinton's memoir What Happened, read aloud by Clinton herself. This was leisure, not business—though she had spent plenty of hours studying the inflections of Clinton's voice in order to master it for Saturday Night Live. “I walk and I listen,” McKinnon told me when we met up this spring. “That's my free time. That's my whole life.” With Clinton, she said, her hope was to create an impression that articulated “the tension between her ambition and really wanting to do something in the world, the self-sacrifice and the self-censure that it takes to do that.”

McKinnon's recurring characters on SNL have that same mix of hard and soft edges, from Sheila Sovage, the drunk trying to hook up before last call, to Ms. Rafferty, the hard-bitten survivor of an alien abduction. But it's her impressions of real people that have produced some of her most unforgettable moments on-screen. In addition to Clinton, she's played Justin Bieber, Kellyanne Conway, Ellen DeGeneres, Robert Durst, Ruth Bader Ginsburg, Angela Merkel, Robert Mueller, Jeff Sessions, and dozens more.

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Even when she's poking fun at her subjects—which is always—she imbues them with an empathy bordering on love. “That's the name of the game,” she said. “You can't be judging someone as you're embodying them. You have to find the point of connection—something you find delightful. Even if you're intending to skewer someone, you also have to find something that you truly like about them. If you're mean, it ain't fun to watch. And if it ain't fun to watch, they turn off the TV.”

There have been moments when that embodiment—to adopt the word that McKinnon frequently uses to describe her impressions—seemed almost to overwhelm her. The week after the 2016 presidential election, as Clinton, she sang a version of Leonard Cohen's “Hallelujah” for SNL's “cold open,” the sketch that kicks off the show, that nearly brought her to tears.

The original plan that night was for each of the female cast members to talk to the camera, one by one, about how she felt after Donald Trump's victory, culminating in McKinnon singing John Lennon's “Imagine.” The prop people went so far as to procure a white piano for her to play. But late in the week, producer Lorne Michaels told me, he decided that approach was too partisan. “In the end, we're a comedy show,” he said. “You can't forget that.”

Instead, Michaels suggested that McKinnon sing “Hallelujah,” solo. “I think she was conflicted about that,” he recalled, “because she was very supportive of it being a group thing.” But if anyone could represent the whole cast in a way they could be proud of, Michaels said, it was McKinnon: “There are very few people who can do this kind of work on the level that she does. It's genius.”