Bring a large pot of lightly salted water to a boil, then cook the thin noodles until al dente. The strands should be tender but still offer a slight resistance when bitten.

While the noodles cook, whisk the eggs, parmesan cheese, parsley, salt, and black pepper in a mixing bowl. Keep whisking until the cheese is distributed throughout the eggs and no streaks remain.

Set a large nonstick skillet over medium heat and melt the butter as the pasta nears doneness. The butter should foam gently and smell warm and nutty without darkening.

Drain the cooked noodles promptly and transfer them directly to the buttered skillet. Steam should rise from the pasta, showing that it is hot enough to help thicken the egg mixture.

Toss the noodles through the butter until every strand looks glossy. The pasta should move freely in the skillet rather than sitting in a clump.

Remove the skillet from direct heat, pour in the egg mixture, and stir rapidly with a flat wooden spoon for less than a minute. The sauce should turn creamy and thick with a shiny sheen as it coats the noodles.

Stop stirring as soon as the coating looks smooth and serve immediately. The finished noodles should look velvety, not dry or dotted with firm egg curds.