Place the flour on a clean dry work surface and make a small well in the center.
Pour the whisked eggs, olive oil, and 1 tbsp salt into the well.
Use a fork to gently incorporate the flour from the edges into the egg mixture until a shaggy dough forms; add a little water only if the dough is too dry to come together.
Knead the dough by hand for about 8–10 minutes until smooth and elastic, or process and then knead: pulse ingredients in a food processor until shaggy, then knead 5–10 minutes until smooth, or knead with a stand mixer and dough hook for about 10 minutes.
Wrap the dough tightly in plastic wrap and refrigerate for at least 30 minutes or up to overnight to rest and firm.
When ready, cut the dough into 1-inch discs and lightly dust with flour.
Roll each disc by hand to desired thinness or pass through a pasta roller, dusting with flour as needed and gradually reducing the roller setting until the sheet is nearly translucent.
Fold or roll the sheet lightly (do not press) and use a sharp knife or pasta cutter to slice into the desired width for fettuccine, linguine, or spaghetti; separate and dust the cut ribbons with flour and arrange in nests or on a drying rack.
Bring a large pot of water to a rolling boil, then add 1 1/2 tbsp salt per pound of pasta; return to a boil.
Add the fresh pasta and cook 1–3 minutes, depending on thickness, until tender.
Drain and toss the pasta with sauce or serve immediately.
