Put the 1 cup (125 g) all-purpose flour on a clean work surface and make a well in the center. Crack the 2 large eggs into the well.
Using a fork, beat the eggs in the well and gradually draw in a little flour from the sides until the eggs start to thicken. Continue incorporating more flour a bit at a time until a shaggy dough forms.
Bring the dough together with your hands. If it is sticky, dust it lightly with more flour. Knead the dough by pushing it away with the heel of your hand, folding it back over itself, giving it a quarter turn, and repeating for about 8 minutes, or until the dough is smooth and elastic.
Shape the dough into a ball, divide it in half, and cover both pieces with a kitchen towel (or plastic wrap) to keep them from drying while you work.
Lightly flour your work surface, rolling pin, and pasta machine (if using). Flatten one half of the dough with a rolling pin into a rectangle thin enough to feed through the widest setting of your pasta machine.
Feed the dough through the widest roller setting. Lightly flour the sheet, fold it into thirds lengthwise, flatten it with the rolling pin, and pass it through the widest setting again. Repeat this flour–fold–flatten process about 10 times total (or until the sheet is smooth and supple).
Thin the sheet by moving the machine to the next narrow setting and passing the sheet through. Continue one setting at a time until the sheet reaches the thickness you want for fettuccine (for many machines this is the second-thinnest setting). If the sheet becomes too long, cut it in half and continue with one piece at a time.
Dust a parchment-lined baking sheet lightly with flour. Lay the finished pasta sheets on the sheet, separating layers with parchment and light flour dusting. Cover with a kitchen towel to prevent drying. Repeat steps 5–7 with the second half of the dough.
Wrap the tray of pasta lightly with plastic wrap (or cover with a towel) and let it rest for at least 10 minutes before cutting so the sheets are easier to handle.
When ready to cut, dust the pasta sheets and cutter with flour. Feed each sheet through a fettuccine cutting attachment or use a sharp knife to cut into fettuccine-width strands. Dust the strands lightly with flour, gather them lengthwise, and form small nested mounds. Transfer the nests to the prepared baking sheet.
If you are not cooking the pasta immediately, cover it tightly with plastic wrap and refrigerate up to 3 days or freeze up to 1 month.
To cook the pasta, bring a large pot of salted water to a rolling boil. Add the fresh fettuccine and cook for 1 1/2–2 minutes, or until al dente. Drain the pasta.
While the pasta cooks or immediately after draining, make the Alfredo sauce: in a small saucepan over low heat, simmer the 1/2 cup (119 ml) heavy cream and 1 tablespoon butter for about 5 minutes, stirring occasionally, until the cream slightly thickens.
Remove the pan from the heat and stir in the 1/2 cup (50 g) grated Parmesan cheese until mostly melted and the sauce is smooth. Season with freshly ground black pepper and salt to taste.
Toss the drained pasta with the Alfredo sauce until evenly coated, then serve immediately.