Bring a pot of lightly salted water to a boil and cook the fresh fettuccine until al dente; reserve 1/4 cup of the pasta water and drain the pasta when ready.
Melt the butter in a medium saucepan over medium-high heat, then add the minced garlic and cook for 2–3 minutes until fragrant but not browned.
Add the heavy cream to the pan and bring 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.
Pour the tempered egg mixture back into the saucepan, whisking constantly; simmer until the sauce is lightly thickened, about 2 minutes.
Whisk the grated Parmigiano-Reggiano into the sauce until melted and smooth, then stir in the lemon zest, grated nutmeg, salt, and 2 tablespoons of the reserved pasta water.
Add the hot drained pasta to the sauce and toss with pasta tongs until thoroughly coated, adding more reserved pasta water a little at a time if needed to reach a silky consistency.
Serve immediately for best texture.