In a Dutch oven, bring 4 quarts water to a boil. Stir in the pasta and 1 tablespoon salt, then cook, stirring occasionally, until just shy of al dente. Reserve about 2 cups of the cooking water, then drain and set aside.

In the same pot over medium-high, cook the oil, garlic and dried basil, stirring constantly, until the garlic is lightly browned, about 2 minutes. Add half of the fresh basil, the chili, tomato paste and ½ teaspoon each salt and black pepper. Cook, stirring often, until the paste browns and slightly sticks to the pot, 1 to 3 minutes. Add 1 cup of the reserved pasta water and scrape up the browned bits, then add the pasta and cook over medium-low, tossing to coat the noodles, until the pasta is al dente, about another 2 minutes; add more reserved water as needed so the pasta is silky and lightly sauced.

Off heat, stir in the remaining basil, then taste and season with salt and pepper. Serve drizzled with additional oil and sprinkled with cheese (if using).