In a medium bowl, whisk together the flour, semolina and salt with a fork.
Create a deep well in the middle of the flour and crack the eggs and egg yolks into this well. Add the olive oil and whisk the eggs with the fork to break the yolks and combine.
As you whisk the eggs, begin gradually pulling in flour from the bottom and sides of the bowl. Don’t rush this step. At first, the eggs will start to look like a slurry and slightly lumpy. Once enough flour has been added, it will start forming a very soft dough and become harder to whisk with a fork. Don’t worry if you haven’t used all the flour.
Turn the dough and any excess flour out onto a clean counter. Begin gently folding the dough on itself, flattening, and folding again. It will be extremely soft at first, then gradually start to firm up. Clean your hands of sticky dough and add a bit more flour to the dough as your knead. Once it’s firm enough to knead, begin kneading the dough. Incorporate more flour as needed to prevent the dough from sticking to you or the counter. Slice into the dough with a paring knife; if you see lots of air bubbles, keep kneading. The dough is kneaded when it forms a smooth elastic ball and has very few air bubbles when sliced.
Clean and dry the mixing bowl. Place the ball of dough inside and cover with a dinner plate or plastic wrap. Rest for at least 30 minutes to allow the flour to hydrate and the gluten to relax.
*At this point, the pasta dough can be refrigerated for up to 24 hours. Let it come back to room temperature before rolling.
Sprinkle a baking sheet generously with semolina flour and place the ball of dough on top (it will stick to the bowl; use a spatula or bowl scraper if necessary). Divide the dough into four equal portions. Dust the portions with flour and cover with a clean dishtowel. Going forward, keep everything well-floured to prevent the pasta from sticking to itself or the pasta roller.
Set your pasta machine to the thickest setting (usually marked “1”). Flatten one piece of dough into a thick disk between your hands and feed it through the pasta roller. Repeat once or twice. Fold this piece of dough into thirds, like folding a letter, and press it between your hands again. With the pasta machine still on the widest setting, feed the pasta crosswise between the rollers. Feed it through once or twice more until smooth. If desired, repeat this folding step. This helps to strengthen the gluten in the flour, giving it a chewier texture when cooked.
Begin changing the settings on your roller to roll the pasta thinner and thinner. Roll the pasta two or three times at each setting, and don’t skip settings (the pasta tends to snag and warp if you do). If the pasta gets too long to be manageable, lay it on a cutting board and slice it in half. Roll the pasta as thin as you like to go. For linguine and fettuccine, I normally go to 6 or 7 on the KitchenAid attachement; for angel hair or stuffed pastas, I go one or two settings thinner.
Cut the long stretch of dough into noodle-length sheets, usually about 12-inches. If making filled pasta or lasagna, proceed with shaping. If cutting into noodles, switch from the pasta roller to the noodle cutter, and run the sheet of pasta through the cutter. Toss the noodles with a little flour to keep them from sticking and gather them into a loose basket. Set this basket on the floured baking sheet and cover with a towel while you finish rolling and cutting the rest of the dough.
To cook the pasta immediately, bring a large pot of water to a boil, salt the water, and cook the pasta until al dente, 4-5 minutes. To dry, lay the pasta over a clothes drying rack, coat hangers, or the back of a chair, and let air dry until completely brittle. Store in an airtight container for several weeks. To freeze, either freeze flat in long noodles or in the basket-shape on a baking sheet until completely frozen. Gather into an airtight container and freeze for up to three months. Dried and frozen noodles may need an extra minute or two to cook.