Bring a large pot of salted water to a boil. Boil the pasta until al dente according to the package instructions.
While the pasta cooks, make the pesto. Add the tomatoes, peppers, hazelnuts, vinegar and chili sauce to the carafe of a food processor. Process until smooth, about 2 minutes. With the machine running, slowly drizzle in the olive oil. Season the mixture with salt and pepper. If the mixture seems too thick, add a splash of pasta water.
Add the cheese, butter, half of the pesto and a ladleful of pasta water to a large bowl. When the pasta is ready, drain and add it directly to the bowl. Stir together, adding more pesto as desired, or more pasta water to reach the desired consistency. Grate more cheese over the pasta and garnish with basil leaves. Serve immediately.
Extra pesto, placed in an airtight container and topped with a layer of olive oil, will keep in the refrigerator up to 2 weeks.