Make the pasta dough: Mound the flour on a work surface and make a large well in the center. Add the eggs to the well and beat with a fork, gradually incorporating flour from the sides until a shaggy dough forms.
Knead the dough: Gather the dough and knead by pushing with the heel of your hand, folding and turning, for about 8 minutes until smooth; add a little flour only if the dough is sticky.
Rest and divide: Divide the dough in two, cover one half with a towel to prevent drying, and keep the other on a lightly floured surface. Prepare parchment-lined baking sheets dusted with flour.
Sheet the dough: Flatten one piece with a rolling pin to fit your pasta machine. Pass it through the widest roller setting several times, flouring and folding between passes until smooth. Then progress through narrower settings one notch at a time until the sheet reaches your desired thickness (about the second-thinnest setting for fettuccine).
Arrange and rest sheets: Dust sheets with flour, layer them on the baking sheet separated by parchment, and cover with a towel. Repeat with the remaining dough and let the sheets rest for at least 10 minutes wrapped to avoid drying out.
Cut into fettuccine: Lightly flour the sheets and feed them through a cutting attachment or use a knife to cut into fettuccine. Dust the cut pasta with flour, gently form into small nests, and transfer to the prepared sheet.
Cook the pasta: Bring a large pot of water to a boil, optionally add a little olive oil, and cook the fresh fettuccine 1½–2 minutes until al dente. Drain the pasta, reserving a little cooking water if desired.
Make the Alfredo sauce: In a saucepan over low heat, simmer the heavy cream and butter until slightly thickened, about 5 minutes. Remove from heat, stir in the grated Parmesan until melted and incorporated, and season with salt and freshly ground black pepper to taste.
Toss and serve: Add the drained pasta to the sauce and toss to coat, adding a splash of reserved pasta water if needed to loosen the sauce. Serve immediately.