Bring 4 quarts of salted water to a rolling boil. Add penne and cook for 8 minutes. Add broccoli florets to the boiling water and continue cooking for 3 minutes until pasta is al dente. Drain and reserve.
Heat olive oil in a large skillet over medium-high heat. Season chicken with half the salt and pepper. Sauté chicken until golden brown and internal temperature reaches 165°F (74°C). Remove chicken from skillet.
In the same skillet over medium heat, melt butter. Add minced garlic and red bell pepper. Sauté for 2 minutes until fragrant and peppers slightly soften.
Pour heavy cream into the skillet. Bring to a light simmer and cook for 5 minutes until liquid reduces slightly.
Reduce heat to low. Whisk in Parmesan cheese gradually until fully incorporated and sauce is smooth. Season with remaining salt and pepper.
Add the cooked penne, broccoli, and chicken to the skillet. Toss continuously over low heat until the sauce coats all ingredients evenly.
Serve immediately at a temperature of approximately 140°F (60°C).