Add the eggs, flour, olive oil, salt and water to the bowl of a stand mixer fitted with the paddle attachment. Mix on low speed until the dough starts to come together, about 3 minutes. Switch to the dough hook attachment and knead until the dough is smooth and soft, about 5 minutes (see note).
Form the dough into a disk and wrap it tightly in a piece of plastic wrap. Refrigerate for at least 30 minutes.
Unwrap your dough and place it on a generously floured surface. Cut the disk into 4 equally-sized pieces and flatten each to about ¼ inch.
With your pasta roller set to the widest setting (0 on KitchenAid), insert the dough through the top. Once it goes through, fold the dough in half. Flour your dough and run it through a second time.
Increase the setting by one step and insert the sheet again, but don't fold it over this time. Repeat this step to your desired thickness. For most pasta varieties this is in the 4-6 range.
Change the attachment to the appropriate cutter. Generously flour the dough, and insert the dough carefully through the top of the attachment. Once you have the pasta cut, toss with a little bit of flour to prevent it from sticking. Set aside, covered with a damp cloth until ready to cook or set out to dry (see drying instructions above).
When ready to cook, bring a large pot of salted water to a boil. Add the pasta and cook until al dente, about 2-4 minutes for fresh pasta and 6-8 minutes for dried.
Serve with your favorite sauce.