Place the two eggs, two yolks, olive oil, garlic, and basil into the Vitamix container and secure the lid. Select Variable 1 (not 1+), turn the machine on, and slowly increase speed to the highest setting. Blend for 20 seconds.
Remove the lid and add the flour and the salt. Place the lid back on and select variable 3. Using the pulse button, pulse 5-6 times. The dough should start to come together. Remove the lid and scrape down the sides as necessary.
Place the lid back on and pulse the dough an additional 5-6 times. Remove the lid and check the dough. If it seems wet and sticks to your finger when touched, sprinkle in a bit more flour, and pulse an additional 2-3 times. The dough should form a ball at this point, but it's also okay if a few pieces don't fully incorporated.
Dump the dough onto a lightly floured work surface and knead for 4 minutes. If the dough feels sticky and clings to your hands, continue to sprinkle in some additional flour and continue kneading. Once the dough is smooth and elastic, tightly cover in plastic wrap and let rest at room temperature for 30 minutes.
Cut dough into four pieces. Working with one piece at a time (leaving the others wrapped), lightly dust the portion with flour and flatten into a disc with your hand. Next, feed the piece of dough through your pasta machine starting on the widest setting. Do this two times. Fold your dough into thirds and feed it through the pasta machine lengthwise. Do this two times as well.
Adjust the machine to the next setting and continue to roll out the pasta dough, 2 passes for each setting, all the way down to the last and narrowest setting. Once on this last setting, the pasta strip will likely become quite long, so go ahead and cut it in half—this will make it easier to handle.
Use the machine's fettuccini attachment to cut the pasta into strips. Dust with flour or semolina if using right away, and lay out onto a baking sheet. Alternatively, you can also dust the pasta and gently form into nests. If not using the pasta right away you can place it on a drying rack and let dry overnight.
Repeat this process for the 3 other pasta portions. If cooking right away, pasta will cook in heavily salted, boiling water in 2-3 minutes. Top with your favorite sauce and enjoy.
Some Additional Pasta Tips
The most important thing to pay attention to when making pasta is to focus on the texture of the pasta dough. If the dough is too sticky you'll end up with one big mess in the end, so let the pasta dough tell you how much flour it needs and add accordingly. The goal is to have a ball of dough that isn't sticky, but is still slightly moist and pliable. Alternatively, if you find your dough to be to dry, add a tiny drop of water until the texture of the dough corrects. I found this article to be very helpful, it even suggested using a water spray bottle if you end up with dry dough, which I found to be genius!
If you don’t have a pasta machine, check out Cooks Illustrated or Bon Appetit for tips on how to make pasta without one.