Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large Dutch oven or pot over medium-high heat. Add the garlic and cook, stirring, until golden, about 30 seconds. Add the onion, bell peppers and 1/2 teaspoon salt. Cook, stirring, until softened, about 5 minutes. Add the tomatoes and cook until softened, about 7 minutes.
Add the chicken, basil, pasta and the reserved cooking water to the tomato sauce. Cook, stirring, until warmed through, about 4 minutes. Drizzle each serving with olive oil and top with basil and the cherry peppers.