Make the dough:
Put the flour and salt in a food processor fitted with the plastic dough blade. Pulse three or four times to blend. With the processor running, pour the olive oil down the feed tube and then slowly pour in the eggs. Continue processing until the dough clumps around the blade, about 10 seconds. It will be a little wet and sticky.
Transfer the dough to a lightly floured surface and knead it by hand until it’s smooth, about 2 minutes. If it seems very sticky and too wet, add a little more flour and continue kneading for 1 more minute. Shape the dough into a ball and dust it lightly with flour. Wrap tightly with plastic and let rest at room temperature for 30 minutes (or refrigerate or freeze).
Roll out the dough:
Tip:Because each machine is different, it’s difficult to say at what exact setting to stop. For lasagne, you want the pasta to be no thicker than 1/16 inch, but not so thin it will tear. It’s usually best to stop a setting or two before the final (thinnest) setting—#5 or #6 on a KitchenAid attachment, #7 or #8 on an Atlas machine.
Unwrap the dough ball and cut it into four equal pieces. (Let refrigerated dough come to room temperature; defrost frozen dough in the fridge and let it come to room temperature.) Take one piece and leave the other three covered in plastic. Flatten the piece of dough with the palm of your hand until it’s about 1/2 inch thick and shape it as close to a rectangle as possible.
Dust the flattened piece lightly with flour and pass it through the machine rollers at the widest setting
Fold it in three, like a letter. Dust the dough with more flour and pass it through again, still folded, starting from a short side. Repeat this step two or three times, folding the dough in three each time and dusting with flour as needed
Then without folding the dough, repeatedly pass it through the machine rollers, reducing the space between the rollers after each pass (dust the dough with flour on both sides before each pass).
When the dough becomes so long that it’s difficult to manage, trim the uneven ends and cut the dough in half crosswise to make two more-manageable lengths of dough. Continue passing each piece through progressively narrower settings. If you're making pasta sheets for lasagne, stop on the second- or third-narrowest setting; the pasta should be slightly translucent but not so thin that it will tear. For other types of pasta, like fettuccine, linguine or ravioli, continue to the thinnest setting.
As you finish rolling each sheet, set it on a lightly floured kitchen cloth and cover with another cloth to prevent the dough from drying out. Or, if not using within an hour or so, layer the sheets between parchment, plastic wrap, or waxed paper to keep them from sticking together and refrigerate or freeze.
The pasta dough can be refrigerated for 24 hours or frozen for up to 1 month, either before or after rolling into sheets.