To make the pasta, place the flour and salt into the bowl of a food processor and pulse to combine. Add the eggs and olive oil and run the food processor until the dough forms.

Turn the dough onto a lightly floured work surface and knead for a couple minutes until the dough is smooth. Form the dough into a round disk and cover with plastic wrap. Allow the dough to rest at room temperature for 30 minutes or refrigerate overnight. (If the dough is refrigerated, make sure to let it come to room temperature before rolling out the pasta.)

(Note: Fresh pasta can be made by hand. Combine the flour and salt. Make a well in the center and add the eggs and olive oil. Whisk the eggs and olive oil and gradually incorporate the flour. Once the dough has formed, knead until it becomes smooth, about 10 minutes.)

While the pasta dough is resting, make the pesto. Mince the garlic cloves in the bowl of a food processor. Add the walnuts, Parmesan, basil, salt and pepper. Puree the mixture. While the motor is running, drizzle the olive oil through the feeder tube until the pesto is fully combined.

Divide the pasta dough into 4 pieces keeping them covered. Working with one piece of dough at a time, flatten slightly and feed through the smooth rollers of a pasta machine starting at the widest setting. Fold the pasta sheet in half each time it is fed through at a narrower setting until the pasta sheet is very thin but does not tear. Place the pasta sheets on a floured work surface to dry for 10 minutes. (Note: If you do not have a pasta machine, you can roll out the sheets of pasta with a rolling pin on a floured work surface.)

Cut the pasta sheets into 12-inch lengths. Sprinkle with flour and roll them up loosely. Using a knife, trim the rough edges and cut into 1/4-inch wide strips. Unroll and place the strips on a flour-dusted sheet pan. Repeat with the remaining dough.

Bring a large saucepan of salted water to a boil. Cook the pasta for 2 minutes until al dente.

Use a slotted spoon to transfer the pasta to a large bowl. Toss with pesto. Garnish with walnuts and Parmesan before serving.