Bring a large pot of salted water to a boil over high heat, and cook the fresh fettuccine pasta until just tender, about 3 to 5 minutes. Drain the pasta, reserving 1/2 cup of the cooking liquid. Return the pasta to the pot.
Heat the olive oil in a large skillet over medium-high heat. Add the onion and season with salt, and pepper. Add the yellow squash and saute until almost tender. Add the lemon zest and continue cooking for 1 more minute.
Add the white wine, and scrape up the pan releasing any dry bits on the bottom. Add the chicken stock, heavy cream, lemon juice, and reserved cooking liquid. Bring to a simmer and continue cooking until the sauce is creamy and coats the back of a spoon. Stir in the grated Parmesan cheese. Season the sauce with salt, and pepper, to taste.
Toss the pasta with the sauce and serve. Garnish with additional Parmesan cheese.
To make fresh pasta dough:
Mound the flour on a clean work surface. Hollow out the center making a well in the middle of the flour with steep sides.
Break the eggs into the well. Add the salt, and olive oil to the hollow center and gently mix together with a fork. Gradually start incorporating the flour by pulling in the flour from the sides of the well. As you incorporate more of the flour, the dough will start to take shape.
With your hands or a bench scraper continue working the dough until it comes together. If the dough is too dry, add a little water; if too wet or sticky, add a little flour.
Begin kneading the dough and keep kneading until it becomes smooth and elastic, about 8 to 10 minutes. At this point, set the dough aside, cover it with plastic, and let it rest for 15 to 20 minutes. You can store the dough in the refrigerator for up to 24 hours, but allow it to return to room temperature before rolling it out.
Divide the pasta dough into 4 even sections. Keep each section covered with plastic wrap or a clean towel while you work with each one. Flour the dough, the rollers of a pasta roller (or your rolling pin), your hands, and the work surface.
If using a pasta machine: Flatten 1 of the of the dough pieces between your hands or with a floured rolling pin until it forms a thick oval disk. Dust the disk, the roller, and your hands with additional flour. Flour a baking sheet to hold the rolled out finished pasta.
With the roller on the widest setting, pass the pasta through the machine's rollers a few times until it is smooth. Fold the dough over into 1/3, and continue to pass through a few more times until the pasta is smooth again. Begin adjusting the pasta machine settings to become thinner, passing the dough through a few times at each setting.
If rolling the pasta by hand: Flatten a dough piece into a thick oval disk with your hands. Flour a baking sheet for the rolled out finished pasta. Place the oval dough disk on a floured work surface, and sprinkle with additional flour. Begin rolling out the dough with a floured rolling pin working from the center of the dough outwards, constantly moving the dough and lifting it to make sure it's not sticking.
Flavor and color variations:
Cook's Note: All color additions should be added to eggs before incorporating into the flour. Keep in mind that you will need to add a little bit of flour to your basic pasta recipe to compensate for the additional liquid added
Spinach Pasta Dough (green hue): Blanch 8 ounces (about 3 cups) spinach leaves. Puree in a food processor or blender until smooth. Press the spinach puree through a small strainer or chinois, removing the liquid before mixing with the eggs. Add 3 tablespoons spinach puree to the eggs before mixing. If the dough feels too sticky, add additional flour until a soft, silky dough is achieved.
Tomato Pasta Dough (orange hue): Add 2 1/2 tablespoons tomato paste to the eggs before incorporating into the flour.
Beet Pasta Dough (pinkish-red hue): Preheat the oven to 400 degrees F. Roast 1 large beet in aluminum foil about 45 minutes, or until tender. Peel the beet and puree in a food processor. Add the beet puree (about 4 tablespoons) to eggs before incorporating into the flour.
Herb Pasta Dough (speckled): Mince 1/2 cup fresh herbs and add to eggs before incorporating into the flour.
Squid Ink Pasta Dough (black hue): Add 1 tablespoon squid ink to the eggs before incorporating into the flour.