Heat a large heavy pot over low, adding 1/2 cup of oil when the pan is hot.
Add garlic and cook, stirring, until softened and fragrant but not browned, about 2 minutes. Add the carrots to sweat them for about 3 minutes.
Increase heat to medium and add the white wine, stirring, for about 2-3 minutes to cook off the alcohol.
Add tomatoes, red pepper flakes, basil, and 1½ tsp. salt. Cook, stirring to coat, until tomatoes begin to burst, about 4 minutes. Smash the tomatoes with the back of a wooden spoon to release liquid, then continue to cook, stirring occasionally, until a chunky, thickened sauce comes together, 10–12 minutes.
Taste and adjust seasoning,
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain pasta, add to pot with sauce, and cook over medium heat, stirring, until coated, 1–2 minutes. Remove from heat and stir in 1 oz. Parmigiano Reggiano.
Divide pasta among bowls. Top with more Parm and fresh basil leaves. Drizzle with good extra virgin olive oil and serve