Cook the pasta in a large pot according to the package instructions. Add the broccoli to the pot for the last minute of cooking. Drain and set aside.
In a large skillet, heat the olive oil over medium heat. Add the chopped onion and cook for 3-4 minutes until golden. Add the minced garlic and cook for a few seconds before adding the chicken strips. Season with salt and pepper and cook for 5-7 minutes until the chicken is browned.
Pour in the cream, grated Parmesan, and pesto. Let it simmer for 2-3 minutes. Add the cooked pasta and broccoli to the skillet, stirring until everything is well combined. Simmer for an additional 2-3 minutes. Remove from heat and garnish with extra Parmesan if desired. Enjoy your delicious Chicken Pesto Pasta!