On a clean work surface, make a mound with the flour and a well in the center. Add the eggs, sourdough starter, salt, and olive oil into the well. Use a fork to whisk and slowly pull flour into the mixture until a shaggy dough forms.
Knead the dough by hand or with a stand mixer and a dough hook attachment for about 8–10 minutes until smooth and elastic. If it feels sticky, dust with a little flour. If it’s dry, add a teaspoon of water at a time. The dough should feel firm and smooth, not sticky. If it becomes too wet, simply dust with a little flour until it comes together. Form into a dough ball, wrap tightly in plastic wrap, and let it rest for at least 30 minutes at room temperature (or refrigerate overnight for deeper flavor and longer fermentation). This rest period allows the gluten to relax and makes it easier to roll thin pasta sheets later.
Cut the dough into 4 equal pieces. Flatten them into a rough rectangular shape.
If using a pasta machine, set it to the widest setting and feed the dough through several times (about 5-8 times), re-rolling until smooth. (With my pasta machine, the Marcato Ampia Device, I start running the dough through setting #1, then run it though the #2 setting, followed by the #3 setting, then the #4 setting. Finally, I run it through the fettuccini setting. This is the thickness we like.) Gradually work down to your desired thickness.
If rolling by hand, use a heavy rolling pin on a floured work surface, turning the dough often to prevent sticking.
Once you’ve rolled out your fresh sourdough pasta sheets, use a pasta cutter, pizza cutter, or sharp knife to cut pasta into your desired pasta shapes, if you are not using a machine.
Dust with flour to prevent sticking, and lay in a single layer on a cookie sheet or baking sheet. You can also hang the noodles on a pasta drying rack if you prefer.
Bring a large pot of water to a rolling boil, then salt generously. Add the fresh pasta, stirring gently. It will cook fast — usually just 2–3 minutes until al dente. Reserve a bit of pasta water to toss with your favorite sauce, helping it cling perfectly. To dry for later, let the pasta air-dry until completely hard, then store in an airtight container or freezer safe bag.