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Cook pasta in a large pot of boiling generously salted water, stirring occasionally, until very al dente, about 2 minutes less than package directions (pasta will finish cooking in the sauce).
Meanwhile, heat half of the butter in a large Dutch oven or other heavy pot over medium until melted. Add lemon slices and cook, stirring often, until softened and bottom of pot is browned in spots, 5–7 minutes. Using tongs, transfer one-third of lemon slices to a plate; set aside.
Just before pasta is al dente, scoop out 2 cups pasta cooking liquid. Add 1½ cups pasta cooking liquid to butter sauce. (This may seem like a lot of liquid, but it will thicken once the remaining ingredients are added.) Add remaining butter a piece at a time, whisking until each piece is incorporated before adding more, until the sauce is emulsified and creamy.
Drain pasta and add to sauce. Cook, stirring often and adding 1 oz. Parmesan a little at a time. Once all of the cheese is added, continue to cook, still stirring, until cheese is melted and sauce is creamy and clings to pasta, about 3 minutes. If sauce looks very thick, add more pasta cooking liquid 1–2 Tbsp. at a time to thin (saucier is ideal as it will thicken as it cools). Remove from heat and sprinkle with an almost ridiculous amount of pepper (about 2 tsp.); toss once more.
Serve pasta topped with reserved lemon rounds and more Parmesan.