In a food processor, pulse the eggs, olive oil, and salt together 8-15 times until the mixture becomes custard-like.
Add about 1 ¼ - 1 ⅓ cups of the flour, then pulse to combine.
Continue adding flour, ¼ cup at a time, until the dough comes together and springs back when you poke it. If the dough still feels quite tacky/sticky, it needs more flour. If the dough starts to come apart and look like little pieces of couscous, gradually add cold water, 1 tablespoon at a time, to reform into a soft dough.
Place the soft dough on a lightly floured board, fold the dough toward you, turn it clockwise by a quarter turn, then repeat this fold and turn approach (it's really just kneading) until you reach the desired smooth and elastic dough stage.
Let the dough ball rest on a lightly floured surface, covered by a bowl to protect it, for at least 30 minutes.
While you're letting the dough rest, start your pasta sauce, which is going to take about 60 minutes to fully develop.
After the resting period, divide your dough ball into 3 equal pieces.
Place your pasta rollers on their widest setting (typically setting #1). Place a small mound of flour on a cutting board or other flat, clean surface.
Lightly brush one side of your dough through the flour and gently pass it through the rollers on this wide setting. Lightly flour the bottom of your dough again, then place it on a clean wooden board or other surface. Fold your dough into thirds lengthwise, then press the air out of the folded dough using your finger tips or knuckles, starting from the bottom and working up to the top of the dough. Repeat this process until you regain the smooth and elastic dough you obtained at the end of the first kneading - this took about 8-12 flour-roll-fold-and-press iterations for us.
When you're ready to move on from kneading with the pasta rollers to stretching, fold your dough one last time so that it is the same width as your rollers. Pass it through the machine on the first setting, then change the setting to number 2. Tap the bottom of the pasta dough sheet through the flour, then pass your dough through the rollers. Decrease the width of the rollers again, flour your dough, and pass it through. Decrease the width of the rollers once more, flour your dough, and pass it through. At this stage, your dough should be ⅛-inch thick. If it's thicker, decrease the width of your rollers again and roll out your dough until it is ⅛-inch thick.
If you have a chitarra, place the dough rectangle on the chitarra and roll over it with a rolling pin to cut the pasta into square ribbons. If you don't have a chitarra, roll the dough rectangle horizontally, then cut into thin ribbons with a sharp knife.
Bring a large pot of cold, salted water to a boil. Optionally, add a pinch of chili powder to the water. Cook the fresh pasta ribbons for 2-3 minutes, until al dente. Drain and assemble immediately.