Heat the oven to 400°F (200°C) and lightly grease a 9-by-13-inch baking dish.

Blanch the broccoli in boiling salted water for 2–3 minutes, until bright green and just tender. Drain thoroughly and pat dry.

Season the chicken pieces with the paprika, Italian seasoning, half of the kosher salt, and the black pepper.

Heat the olive oil in a large ovenproof skillet over medium-high heat. Cook the chicken for 5–7 minutes, stirring occasionally, until lightly browned and nearly cooked through, then transfer it to the prepared baking dish if needed.

Reduce the skillet heat to medium and add the butter. Cook the minced garlic for about 30 seconds, just until fragrant.

Add the heavy cream, chicken broth, and cubed cream cheese. Stir for 2–3 minutes, until the cream cheese melts and the sauce becomes smooth.

Stir in the Parmesan and half of the shredded mozzarella. Add the remaining salt and red-pepper flakes if using, tasting carefully before adding more seasoning.

Add the broccoli and chicken to the sauce and mix gently so the florets remain intact. Spread the mixture into the baking dish, or keep it in the ovenproof skillet.

Top the casserole with the remaining mozzarella and bake for 15–20 minutes, until bubbling around the edges and browned on top. Broil for 1–2 minutes for a deeper golden crust, watching closely.

Let the casserole rest for 5 minutes before serving. Check that the thickest chicken pieces have reached an internal temperature of 165°F (74°C).