In the bowl of a food processor or in a large bowl, add flour, tapioca starch, and salt. Pulse to combine.
Add eggs, egg yolks, and olive oil and pulse to combine. Slowly drizzle in water just until dough starts to come together on top of the blade. If mixing in bowl, add water until dough feels slightly tacky, not dry at all, but not really wet.
Knead until smooth on flour surface and wrap in plastic wrap. Allow to sit for 30 minutes.
Working with half the dough at a time, rewrap the other half in plastic wrap and set aside. Flatten the dough into about 1/4-inch thick. Run it through a pasta sheeter 2-3 times on the 1st setting to smooth it out, folding it over itself before putting it back through. Turn the knob to the 2nd setting and run it through once, then the 3rd, 4th, and 5th, or to as thin as you'd like (I stop after one time on setting 5). Alternatively, roll the dough on a well-floured surface with a rolling pin to desired thin.
Use the spaghetti cutter or fettuccine cutter to cut the dough. Alternatively, roll the dough into a log and cut with a knife to desired thickness.
Cook the pasta in a large pot of salted water until al dente, about 3 minutes, depending on the size and thickness of your pasta.
Store the pasta in the refrigerator, covered, for up to 24 hours. After that, it's best kept in the freezer, well wrapped, until ready to use. Cook from frozen.