Finely grate the Parmigiano Reggiano and cut the unsalted butter into small pieces; set both aside. Place a large heatproof serving bowl and four plates in the sink and fill with very hot tap water to warm them.
Bring 4 quarts of water to a boil in a large pot. Add the fine sea salt and return to a rolling boil.
Add the fresh egg fettuccine and cook until just al dente, 2–3 minutes, with the strands still springy at the center.
Reserve 1.5 cups of the starchy cooking water. Drain the pasta quickly. Empty the warmed serving bowl and dry it.
Add half the unsalted butter and 1/4 cup of the reserved pasta water to the warm bowl. Add the hot fettuccine and toss to begin melting the butter.
Sprinkle in about half of the Parmigiano Reggiano while tossing vigorously, adding splashes of pasta water as needed to form a glossy, creamy emulsion that coats the strands.
Add the remaining butter and then the remaining cheese in two additions, tossing constantly; add more pasta water as needed until the sauce is silky and clings to the noodles, 1–2 minutes. The sauce should look creamy but not oily or clumpy.