Cook pasta in large pot according to package directions until 1 minute before done (al dente). Add broccoli to pasta water 1 minute before draining. Drain both together and set aside.
While pasta cooks, heat olive oil in large pan over medium-high heat. Add minced onion and cook 3-4 minutes until softened and golden.
Add garlic and cook 30 seconds until fragrant.
Season chicken strips with salt and pepper. Add to pan and cook 5-7 minutes, stirring occasionally, until browned and cooked through.
Reduce heat to medium. Add cream, parmesan, pesto, salt, and pepper to pan. Stir to combine and simmer 2-3 minutes until sauce thickens slightly.
Add cooked pasta and broccoli to the sauce. Toss everything together until pasta is well coated. Simmer 2-3 minutes more.
Remove from heat. Garnish with additional parmesan if desired and serve immediately.