Begin by placing the flour in a medium-sized mixing bowl. In a separate small bowl, combine the sourdough starter, eggs, olive oil, and salt, and then pour this mixture over the flour. Use a fork to blend the ingredients until they form large clumps. Gradually add one teaspoon of water at a time to the dough, using your hands to incorporate it until the dough begins to come together. Take note that the dough should not be sticky. If you find that too much water has been added, lightly dust the dough with a small amount of flour.
Turn the dough onto a clean work surface and proceed to knead it by hand for approximately 5 to 10 minutes until it reaches a smooth and elastic texture. Shape the dough into a disc and tightly wrap it with plastic wrap. Let it rest on the counter for a duration of 1 hour. Alternatively, if you prefer a longer fermentation process, you may allow it to rest for up to 24 hours at room temperature or in the fridge.
Divide the disc into 4 equal portions while ensuring they remain wrapped in plastic to avoid drying out. Take one portion of dough at a time and use a rolling pin or a pasta machine to roll out the dough to your preferred thickness. Remember to generously sprinkle flour on both sides of the dough throughout the rolling process to prevent it from sticking. (Refer to the notes below for recommended thickness guidelines.)
After rolling out the pasta sheets, proceed to cut them into your desired shape using a sharp knife, pizza cutter, bench scraper, or pasta cutter. Place the cut pasta on a cutting board or cookie sheet, or hang it on a drying rack to rest for approximately 30 minutes before boiling it or incorporating it into your chosen recipe.