Bring a large pot of salted water to a boil. Add the pasta and cook according to package directions; add the broccoli for the last 2 minutes. Drain and set aside.

Heat the olive oil in a large skillet over medium-high heat. Add chicken pieces in a single layer and let them sear until golden, about 4–5 minutes, stirring once.

Add the minced garlic and cook for about 30 seconds until fragrant, being careful not to burn it.

Sprinkle the flour over the chicken and stir to coat until it forms a light paste.

Slowly whisk in the milk and chicken broth, scraping any brown bits from the pan. Lower the heat to medium and simmer until the sauce thickens, about 3–4 minutes.

Once thickened, stir in salt, pepper, cheddar, and Parmesan until the cheese melts and the sauce is smooth and glossy.

Toss the cooked pasta and broccoli into the skillet, stirring until everything is combined and heated through. Serve hot.