Bring a large pot of salted water to a boil. Cook linguine until al dente according to package instructions. Reserve about 1/2 cup of pasta water, then drain.
While the pasta is cooking, whisk together the egg yolks, heavy cream, lemon zest, and a good pinch of black pepper in a bowl. Set aside.
In a large skillet over medium-low heat, melt the butter. Once melted, stir in the lemon juice. Let it bubble slightly for about 30 seconds to meld the flavor.
Remove the skillet from heat and slowly whisk in the cream and egg yolk mixture. Stir constantly to avoid scrambling the eggs.
Add the drained linguine to the skillet and toss everything together quickly, adding reserved pasta water a little at a time until the sauce reaches a silky consistency.
Stir in the grated Parmigiano-Reggiano. Taste and adjust salt and pepper. If desired, garnish with chopped parsley or basil.
Plate the pasta and serve while hot. Optionally top with extra zest or a sprinkle of cheese.