Pulse tomatoes and their juice in food processor until finely chopped, about 10 pulses.
Heat oil, pepperoncini, tomato paste, garlic, pepper flakes, anchovies, paprika, ½ teaspoon salt, and ½ teaspoon pepper in medium saucepan over medium-low heat, stirring occasionally, until deep red in color, 7 to 8 minutes.
Add tomatoes and Pecorino and bring to simmer. Cook, stirring occasionally, until thickened, about 20 minutes.
Bring 4 quarts water to boil in large pot. Add pasta and 1 tablespoon salt and cook, stirring often, until al dente. Reserve ½ cup cooking water, then drain pasta and return it to pot.
Add sauce and toss to combine, adjusting consistency with reserved cooking water as needed. Season with salt and pepper to taste. Serve, passing extra Pecorino separately.
