Place flour and salt in the bowl of a food processor fitted with the metal blade and pulse to combine.
With the machine running, add eggs one at a time and blend after each addition. Add olive oil and process until dough looks like a coarse meal that comes together when pressed into a ball, about 10 seconds. Do not over-process. If the dough seems excessively sticky, add a little more flour, 1 tablespoon at a time, processing just until incorporated.
Lightly dust work surface and hands with flour. Turn dough out of food processor and knead until elastic and smooth, and no longer sticky, 5 to 10 minutes.
Shape dough into a disk and cover with plastic wrap or an overturned bowl. Let rest for at least 30 minutes at room temperature (or up to 1 day in refrigerator) before rolling. If the dough has been refrigerated, let stand at room temperature for about an hour before rolling and shaping.
Cut the dough into 6 to 8 pieces. Flatten it into small rectangles to fit through a pasta machine.
Keeping the dough lightly floured, roll the pieces through a pasta machine set at its widest opening. Keep rolling pasta through machine as progressively narrower settings, forming strips.
Lay pasta strips on a lightly floured surface and keep covered with towels. One at a time, fold strips into a flat roll and slice crosswise into ribbons. Unfurl ribbons and transfer to a floured baking sheet. Slice lengthwise into long ribbons, about 1 1/2 inches wide, using a sharp knife or pizza cutter.
Cook pasta in boiling salted water for 2 to 4 minutes, until al dente. Drain and serve immediately.