Put the pesto in a large bowl.
Cook the pasta according to package instructions. Drain the pasta and reserve 1/3 cup of the pasta water. Whisk the pasta water into the pesto.
While the pasta is cooking, heat the oil in a large skillet over medium-high heat. Add the garlic and cook, stirring until soft and fragrant, about 30 seconds. Add the chicken, peppers, and olives and cook, stirring until hot, about 3 minutes. Season with salt and pepper.
Add the pasta and cheese to the pesto and toss to combine. Add the chicken mixture and combine. Divide the pasta among 4 bowls and serve immediately. Pass Parmesan at the table.
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