Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until tender yet firm to the bite, about 12 minutes. Drain. Drizzle with 1 teaspoon olive oil; toss to coat.
Heat remaining oil in a large, non-stick skillet over medium heat. Add carrots, onions, and celery; cook and stir about 1 minute. Add peas and bell pepper. Cook until the vegetables have softened, 3 to 5 minutes. Add the cooked spaghetti; toss to mix evenly.
Reduce heat to medium-low. Slowly pour in beaten eggs. Tilt the pan in all directions so the eggs reach edges of pan. Season with ground black pepper. Cover; cook until bottom has cooked and the upper edges have set, 6 to 8 minutes.
Remove cover. Place a large plate over skillet and flip it over. Slide the flipped frittata back into skillet to cook the other side, about 4 more minutes.
Transfer frittata to a warm serving platter; allow to rest 10 minutes.
Top the frittata with warmed sauce and grated cheese.