In a large pot or Dutch oven, combine onion, garlic, carrots, celery, and tomatoes, and drizzle in the olive oil. Chop half of the basil leaves and add them to the pot. Bring the sauce to a simmer for 45 minutes, stirring occasionally to break up the tomatoes.
As the sauce simmers, prepare a 4-quart pot of salted boiling water to cook the pasta.
Take the remaining basil and use scissors to cut the leaves into small pieces. This will preserve the color and appearance of the herb.
Once the diced vegetables are tender, turn off the heat and let the sauce rest for 15 minutes.
Cook the pasta until al dente, then drain and rinse the pasta with lukewarm water. Taste the sauce and add salt to taste if desired.
To serve, top a plate of pasta generously with sauce, then finish with grated Parmigiano-Reggiano cheese and the remaining cut basil leaves.
