Bring a large pot of heavily salted water to a boil. Add the spaghetti and cook according to the package directions. Reserve about 1 cup of the starchy cooking water. Drain the spaghetti, place back into the hot pot, and drizzle with 1 tablespoon of the olive oil. Toss gently to coat the pasta with the oil.
While the spaghetti is cooking, make the pesto. Bring a pot of salted water to a boil. Add the peas and cook for 2-3 minutes or until cooked through. Drain well. Reserve a small handful of the peas for garnish.
Place the garlic in a food processor and chop finely. Add the cooked peas, pine nuts, Parmesan cheese, ½ teaspoon salt, and ½ teaspoon pepper. Process until everything is finely chopped.
With the motor running, drizzle in the remaining 1/3 cup olive oil and process until smooth and fully combined. Add the pesto to the spaghetti along with some of the starchy pasta water. Stir to coat, adding more water a little at a time to thin the sauce if needed.
Garnish with the reserved peas and more Parmesan cheese if desired.