Step 1: Cook the pasta according to package directions in salted water until al dente. Reserve about 1 cup of pasta water before draining. Set aside.
Step 2: While the pasta cooks, heat olive oil in a large skillet or Dutch oven over medium-high heat. Add the chicken pieces and season with salt and pepper. Cook, stirring occasionally, until browned and cooked through, about 5-7 minutes. Remove chicken from the skillet and set aside.
Step 3: Add the broccoli florets to the same skillet (add a little more olive oil if needed). Sauté for 3-5 minutes until tender-crisp and bright green. Add the minced garlic and red pepper flakes (if using) and cook for 1 minute more until fragrant.
Step 4: Pour in the chicken broth, scraping up any browned bits from the bottom of the pan. Bring to a simmer, then reduce heat to medium. Stir in the heavy cream and Parmesan cheese. Season with salt and pepper to taste.
Step 5: Return the cooked chicken and drained pasta to the skillet with the sauce. Toss to combine, ensuring everything is well coated. If the sauce is too thick, add a splash or two of the reserved pasta water until it reaches your desired consistency.
Step 6: Serve immediately, garnished with extra Parmesan cheese and fresh parsley, if desired.