Bring a large pot of salted water to a boil. Add the penne and cook until al dente. During the last 3 minutes of cooking, add the broccoli florets to the pasta water. Before draining, reserve about 1/2 cup of the pasta water. Drain the pasta and broccoli.
Meanwhile, season the chicken pieces with salt, pepper, and Italian seasoning. Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook until browned and cooked through, about 6-8 minutes. Remove the chicken and set aside.
In the same skillet, add the minced garlic and sauté until fragrant, about 30 seconds. Pour in the heavy cream and chicken broth. Bring the sauce to a simmer, stirring occasionally, and cook until slightly thickened, about 3-4 minutes.
Stir the Parmesan cheese into the cream sauce until melted. Season with red pepper flakes, salt, and pepper to taste.
Add the cooked penne, broccoli, and chicken to the skillet with the sauce. Toss everything together until well coated, adding some reserved pasta water a little at a time if needed to reach your desired sauce consistency. Drizzle with lemon juice and toss again.
Serve the chicken and broccoli pasta hot, garnished with extra Parmesan cheese and red pepper flakes if desired. Enjoy!