Bring a large pot of salted water to a boil. Cook the pasta according to package directions, but remove it 1 minute before the suggested al dente time. During the last 2 minutes of cooking, stir in the broccoli florets. Drain the pasta and broccoli together and set aside.

In a medium bowl, toss the chicken pieces with paprika, garlic powder, onion powder, and a generous pinch of salt and pepper. Heat the olive oil in a large oven-safe skillet over medium-high heat. Add the chicken and cook, stirring occasionally, until just no longer pink on the outside, about 3 to 4 minutes.

Pour the chicken broth into the skillet and bring to a simmer. Add the cream cheese cubes and stir until mostly melted and smooth (small specks are okay). Stir in 3/4 cup of the mozzarella and the cheddar cheese until melted and the sauce thickens slightly.

Turn off the heat. Gently fold the cooked pasta and broccoli into the sauce until evenly coated. Transfer the mixture to a 9x13-inch baking dish if your skillet isn't oven-safe. Cover tightly with foil and bake in a preheated 375°F oven for 15 minutes.

Remove the foil and sprinkle the remaining 3/4 cup mozzarella and the Parmesan over the top. Return to the oven, uncovered, and bake for another 3 to 5 minutes, until the cheese is melted and bubbly. Let the dish rest for 5 minutes before serving to allow the sauce to set.