Cook pasta according to package instructions until al dente. Drain and set aside.
Steam or lightly boil broccoli until tender-crisp. Drain and set aside.
In a large skillet, heat olive oil over medium heat. Add garlic and cook until fragrant.
Whisk in flour to form a roux, then slowly pour in milk and chicken broth, whisking until smooth.
Cook sauce until it thickens slightly, then stir in Parmesan cheese.
Add sliced chicken and broccoli to the sauce, then toss in cooked pasta.
Season with salt and pepper, garnish with parsley, and serve warm.