Add flour to the bowl of your stand mixer. Make a well in the middle and add the eggs, salt and olive oil. Then, using a fork, break the egg yolks and begin to mix. Add water in small increments until the dough comes together. Depending on the size of your eggs, the dough may need more or less water, so start slow. Continue to add water until the dough is no longer shaggy. Then, attach the dough hook to your stand mixer and knead on medium-low speed for five to six minutes.
Cover the dough tightly with plastic wrap and let it rest for thirty minutes. Meanwhile, set up your drying racks and your pasta machine. Set the rollers of the pasta machine to its widest setting.
Using a pastry cutter, or knife, cut off an egg sized chunk of dough and cover the remaining dough so it doesn't dry out. Dust the dough with flour so it doesn't stick to the pasta machine, flatten it with your hands and run it through the widest setting on the roller.
Change the setting to the next widest setting and repeat. If the dough sticks to the rollers, or it looks pitted as it comes out the bottom or the roller, dust with flour. Continue this process until the dough is about 1/16" thick. On my kitchen aid attachment, it's number 4.
Cut the pasta sheets into squares about 5" each and lay them on a well floured surface, separated so they don't stick together. Continue rolling and cutting until you have used all the dough. At this point, the pasta can either be cooked immediately, or left to dry.
To make the pesto, add the basil, salt, garlic, pine nuts, ricotta and yogurt to a food processor and pulse until ingredients are combined. Stream in the olive oil and blitz until well blended.
Next, set a large pot of water to boil and salt generously.
Boil the pasta sheets, 2 or 3 at a time, for about 30 seconds to a minute. Remove them from the water using a slotted spoon or kitchen spider and set aside.
To serve, smear pesto on the bottom of a plate, place a sheet of pasta on top, and continue layering pasta sheet and pesto until you have a fair sized portion, finishing the last sheet off with a spoonful of pesto. Add a pinch of grated Parmigiano Reggiano and then press the sides of the pasta sheets, creating a bump in the pasta so the pesto runs down the sides of the pasta. Serve immediately.
