Bring a large pot of lightly salted water to a boil and cook the fresh fettuccine until al dente; reserve 1/4 cup of the pasta cooking water, then drain, keeping pasta ready to add to the sauce.
Melt the butter in a medium saucepan over medium-high heat, then add the minced garlic and cook, stirring, for 2–3 minutes until fragrant but not browned.
Add the heavy cream to the saucepan and bring it to a gentle simmer.
In a small bowl, whisk the 2 egg yolks and temper them by slowly whisking in 1/4 cup of the hot cream until smooth, then whisk the tempered yolks back into the pot while whisking constantly.
Simmer the sauce, stirring, until it thickens lightly, about 2 minutes, then whisk in the grated Parmigiano-Reggiano until melted and smooth.
Stir in the lemon zest, salt, and freshly grated nutmeg, and add 2 tablespoons of the reserved pasta water to loosen the sauce as needed.
Add the hot drained fettuccine directly to the sauce and toss with tongs or a spoon until the pasta is thoroughly coated, adding a little more reserved pasta water if necessary for a silky consistency.
Serve immediately for best texture.
