Make the pesto: Place basil, spinach, pine nuts, garlic, and cheese in a food processor. With food processor running, slowly pour olive oil through top. Process pesto until completely smooth, scraping down sides if necessary. Season with salt and pepper to taste. Set pesto aside.
In a large skillet or Dutch oven, saute chicken, broccoli, onion, and garlic in oil over medium-high heat until chicken is cooked and broccoli is tender, about 8-10 minutes.
Reduce heat to medium and add cooked pasta, pesto, and reserved 1/2 cup pasta water to pan and stir to combine. Cook over medium heat until pasta is heated through. Season pasta with salt and pepper to taste and top with Parmesan cheese. Enjoy!