Cook spaghetti according to the package directions.
Meanwhile, heat the oil in a skillet over medium heat. Sauté the green pepper, scallions, and crushed red pepper (if desired) until tender but not brown, 3 to 4 minutes.
In a medium bowl, whisk the eggs with the Parmesan and salt; add them to the skillet with the spaghetti and basil. Toss gently over heat until eggs are set.
Serve with grated Parmesan and crusty bread.