In a 4-quart saucepan, bring 3 quarts of water to a boil over high heat.
While the water heats, chop onion, mince garlic, and cut broccoli into tiny florets. Set aside.
In a large nonstick pan, heat olive oil until very hot. Add the chicken. Cook and stir until lightly browned on both sides, about 5–8 minutes. Place chicken on a clean plate, and cover to keep warm.
Coat pan with cooking spray. Over medium heat, cook and stir the onion for about 3 minutes. Add the garlic and broccoli. Cook and stir for 2 more minutes.
Return the chicken to the pan. Add entire jar of pasta sauce, cayenne pepper, and salt. Gently mix to blend ingredients. Cover. Simmer until chicken and vegetables are warmed through, about 4 minutes.
Drop pasta into boiling water. Cook according to package directions for the shortest recommended time, about 2 minutes. Drain.
Divide pasta among four dinner plates (about 1 cup each). Top each with one-fourth of the chicken and sauce mixture. Serve immediately.