Cook the fettuccine in a gallon of water at a rolling boil with a tablespoon of salt dissolved in it. Cook only until very flexible, but still firm. Drain and keep warm. Reserve 1/4 cup of the water.
In a large skillet, melt the butter, then whisk in the half-and-half. Bring it to a light boil over medium heat until blended completely. Whisk in the reserved pasta water, then turn off the heat.
Add the hot pasta to and sprinkle on the Parmesan cheese. With two forks, toss the pasta with the sauce until the cheese is blended in. If you like, add an egg yolk at this point and toss quickly to blend.