Cook pasta according to package instructions; drain and set aside.
Heat olive oil in a large skillet over medium heat. Add chicken slices, season with salt and pepper, and cook until golden and cooked through. Remove from skillet and set aside.
Steam broccoli florets until crisp-tender, about 4-5 minutes; set aside.
In the same skillet, add minced garlic and sauté for 1 minute until fragrant.
Sprinkle flour over garlic and stir to create a roux; cook for 1-2 minutes.
Gradually whisk in chicken broth and milk, stirring constantly until sauce thickens.
Remove from heat and stir in Parmesan cheese until melted and smooth.
Return chicken and broccoli to the skillet; toss to coat in sauce.
Add cooked pasta and gently mix everything together.
Serve warm, garnished with extra Parmesan if desired.