Bring a large pot of heavily salted water to a boil. Add pasta and cook according to package directions until al dente. Before draining, reserve about 1 cup of the starchy pasta water. Drain the pasta and set aside. While the pasta cooks, shred the chicken, mince the garlic, and grate the Parmesan.
Steam the broccoli until tender-crisp. You can use a steamer basket over simmering water for 3-4 minutes, or place florets in a microwave-safe bowl with 2-3 tablespoons of water, cover, and microwave on high for 2-3 minutes. Set aside.
In a large skillet or Dutch oven, melt the butter over medium heat. Add the minced garlic and cook for 30-60 seconds until fragrant, being careful not to burn it. Whisk in the flour and cook for 1 minute, stirring constantly, to form a roux.
Slowly whisk in the chicken broth until smooth, then whisk in the heavy cream. Bring the sauce to a gentle simmer and cook for 2-3 minutes until it thickens slightly. Remove from heat and stir in the grated Parmesan cheese until fully melted and the sauce is velvety. Season with salt and pepper.
Add the drained pasta, shredded chicken, and steamed broccoli to the skillet with the sauce. Toss gently to coat everything evenly. If the sauce is too thick, stir in the reserved pasta water, a tablespoon at a time, until it reaches your desired consistency. Serve immediately, topped with extra Parmesan cheese and fresh black pepper.