Process the flour, eggs, egg yolks and olive oil in a food processor for about 45 seconds, until it forms a cohesive ball. The dough ball should feel just slightly tacky to the touch. If the dough feels sticky, you can add 1 tablespoon of flour at a time (up to 1/4 cup). If the dough does not come together, you can add up to 1 tablespoon of water, 1 teaspoon at a time. Pulse the processor for an additional 30 seconds after you have made your adjustments.
Knead the dough ball on a dry surface for 1 to 2 minutes and bring it into a cylinder shape that is about 6 inches long. Wrap the dough tightly with plastic wrap and set aside for a minimum of 1 hour and up to 4 hours to relax the gluten before the rolling the dough out. You can refrigerate the dough before resting if you need longer, then take it out of the refrigerator and rest at room temperature for 1 to 4 hours.
Cut the dough cylinder crosswise into 6 equal pieces. Working with 1 piece of dough (and covering the rest with the plastic wrap), dust with flour, then press it into a square.
Run the dough through the pasta roller at the largest setting. Fold the sheet over into thirds, then rotate 90 degrees and run it through the pasta roller again at the largest setting.
Continue to run the pasta sheet through the roller at a smaller setting each time until the desired thickness is reached. If the dough starts to stick, you can dust with more flour.
Place the rolled pasta sheets on kitchen towels and leave uncovered for about 15 minutes, or until the edges start to dry ever so slightly.
Cut the pasta sheets to the desired noodle or use to make ravioli.
Cook the noodles within the hour in heavily salted water for about 3 minutes.
This recipe is based on the America's Test Kitchen's “Pasta without a Machine” recipe ingredients, though I do use a pasta roller to roll out the dough.