Bring a large pot of lightly salted water to a boil and cook the fresh fettuccine until al dente; stir in a little olive oil and reserve 1/4 cup of the pasta cooking water before draining.
Melt the butter in a medium saucepan over medium-high heat, add the minced garlic, and cook for 2–3 minutes until fragrant but not browned.
Pour in the heavy cream and bring the mixture to a gentle simmer.
Whisk the egg yolks in a small bowl, then slowly whisk in 1/4 cup of the hot cream to temper the yolks.
Whisk the tempered yolk mixture back into the saucepan and cook, whisking constantly, until the sauce is lightly thickened, about 2 minutes.
Stir the freshly grated Parmigiano-Reggiano into the sauce until melted and smooth, then add the lemon zest, salt, freshly grated nutmeg, and 2 tablespoons of the reserved pasta water.
Transfer the hot, drained fettuccine directly into the sauce and toss with tongs until evenly coated, adding more reserved pasta water as needed to achieve a silky consistency.
Serve immediately for best texture.