Whisk together the mayonnaise, basil, parsley, sour cream, lemon juice, 1/2 teaspoon salt and a few grinds of pepper in a large bowl.
Heat a grill or grill pan over medium-high heat until hot; brush with the oil. Season the chicken breast with salt and pepper. Grill the chicken, turning once, until well-marked and just cooked through, about 7 minutes per side. Transfer to a cutting board and let rest for 5 minutes, then cut into 1/2-inch pieces.
Meanwhile, cook the broccoli in the boiling water until crisp-tender, about 3 minutes. Transfer the broccoli to a colander with a slotted spoon and run under cold water to stop the cooking process. Add the broccoli to the bowl with the dressing.
Add the pasta to the boiling water and cook according to the package directions. Drain and rinse under cold water to cool; add to the bowl.
Add the chicken and red peppers to the bowl and toss to combine. Season with salt and pepper. The pasta salad is best if served right away.
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