Cook pasta according to the package. Reserve 1/2 cup of the pasta water, then drain and set aside. Sprinkle some olive oil over the noodles so they don't stick together.
In a large skillet, add 1 tablespoon olive oil and bring to a medium high heat. Sauté broccoli for 3-5 minutes, stirring constantly and slightly tender. Remove to a plate and set aside.
In the same skillet, add remaining 2 tablespoons olive oil over medium high heat. Add garlic and cook until fragrant, about 1 minute. Whisk in flour and cook an additional minute.
Slowly whisk in the chicken broth 1/2 cup at a time, making sure to remove all clumps. Once the chicken broth is added, slowly mix in the milk, greek yogurt, salt and pepper.
Turn heat down to a simmer (you don't want the greek yogurt to curdle) and let the sauce simmer until thickened, about 5 minutes.
Add the sauce, cooked chicken and broccoli to the pot of cooked pasta. Toss to coat entirely. Sprinkle with mozzarella cheese and cover for 2 minutes.
If the pasta is to thick, add some of the reserved pasta water, if needed.
Serve and enjoy!