Add flour and salt to the bowl of a food processor; pulse a few times to combine.
In a medium bowl, mix eggs and olive oil with a fork until just combined. Pour over flour mixture in food processor. Pulse several times until no dry flour remains. Process for about 1 minute, or until dough has gathered into a rough ball then broken into small, bead-like fragments and gathered into a ball once or twice more. (This will both mix and knead the pasta dough.) If the dough gets stuck at any time, turn off the food processor and nudge the dough slightly and continue to process.
Remove the pasta dough from the food processor and shape with your hands into a smooth ball and cover with plastic wrap. Let rest for 30-60 minutes at room temperature. Roll out immediately or refrigerate for up to 1 day (recipe note).
Plan to use your fresh pasta dough within 1 day as the eggs will begin to oxidize and change color if refrigerated for longer. The same goes for fresh pasta noodles - you'll want to use or freeze within 1 day and use within 1 month.