Make the pasta dough: mound the flour on a clean work surface and form a large well in the center. Crack the eggs into the well and beat with a fork, gradually incorporating flour from the inside edge until a shaggy dough forms. If sticky, add a little more flour.
Knead the dough: knead by pressing with the heel of your hand, folding, turning, and repeating for about 8 minutes until smooth and elastic. Cover and let rest briefly while you prepare equipment.
Roll the dough: divide dough in half. On a lightly floured surface, flatten one piece with a rolling pin enough to feed through the widest setting of your pasta machine. Pass through the widest setting several times, folding and flouring between passes, until smooth (about 10 passes).
Thin the sheets: progressively reduce the roller setting, passing the sheet through each setting until you reach your desired thickness for fettuccine (second thinnest setting recommended). Cut sheets in half if they become too long.
Prepare and dry the sheets: dust a parchment-lined baking sheet with flour and transfer rolled sheets, separating layers with parchment and flour. Cover with a towel and let rest at least 10 minutes before cutting.
Cut and nest the pasta: feed each sheet through the fettuccine cutter, dust lightly with flour, gather lengthwise into a bundle, then twist into small nested mounds and place on the prepared sheet.
Cook the pasta: bring a large pot of water to a rolling boil, add a drizzle of olive oil if desired, and add the fresh fettuccine. Cook 1½–2 minutes until al dente. Drain and reserve a little pasta water if needed.
Make the Alfredo sauce: in a saucepan over low heat, simmer the heavy cream and butter together until slightly thickened, about 5 minutes. Remove from heat and stir in the grated Parmesan until mostly melted and combined.
Toss and season: add the drained pasta to the sauce and toss to coat, adding a splash of reserved pasta water if needed to loosen. Season with salt and freshly ground black pepper to taste, then serve immediately.
