Add the flour to a clean work surface, make a well in the center, and whisk in the eggs and yolks with a fork, gradually pulling in flour from the edges.
Once it becomes too thick for a fork, switch to your hands and knead until the dough comes together, then keep kneading for about 8 to 10 minutes until smooth and elastic.
Wrap the dough in plastic wrap and let it rest at room temperature for at least 1 hour so it becomes easier to roll out.
Cut the dough into 3 equal pieces, roll one into a rectangle, and pass it through a pasta roller starting at the widest setting; fold and repeat until smooth, then continue thinning it to about 1/8 inch thick.
Dust the rolled pasta sheets with flour, cut them using a fettuccine cutter, and lay them on a floured tray, folding as needed and covering with a towel.
Bring a large pot of salted water to a boil and cook the pasta for 1 minute, stirring to keep it from sticking—fresh pasta cooks fast, so don’t walk away.
While the water heats, melt butter in a large skillet over medium heat, add garlic, and cook for about 1 minute until fragrant and lightly golden.
Pour in the half and half, heat until it’s steaming (not boiling), then whisk in half the Parmesan along with salt and pepper until smooth.
Add the drained pasta and ½ cup of pasta water to the sauce, simmer for a minute or two, then toss in the rest of the cheese and stir until everything is coated and glossy.