Put the flour and salt in the bowl of a food processor and pulse just to combine. Combine the eggs, olive oil and 1/4 cup cold water in a spouted measuring cup. With the machine running, pour the wet ingredients into the flour and process until the dough begins to form a ball on the blade. If the dough is still too crumbly, add more water a teaspoon at a time until the dough forms a ball. Process to knead and smooth the dough, about 20 seconds.
Dump the dough onto a floured work surface and knead a few times to get a smooth ball. Wrap the dough in plastic wrap and let rest at room temperature for 30 minutes or up to overnight in the refrigerator (bring to room temperature before rolling, if refrigerated).
To roll the dough: Cut the dough into 4 pieces. Keep the pieces covered until you are ready to roll them. Form 1 piece of dough into a rough rectangle shape. Roll and stretch it as thin as possible; it should measure approximately 10-by-12 inches. Keeping the dough well floured, roll it inward from 2 opposite sides, jelly-roll style. Cut the dough at 1/2- inch intervals and unfurl each piece into separate strands. Form the pasta into loose nests and rest on a floured baking sheet. Repeat with the remaining pieces of pasta.
To cook, boil the pasta in generously salted water until al dente, about 2 minutes after the water returns to a boil. Toss with the sauce of your choice.