Preheat oven to 375°F (190°C). Cook pasta according to package instructions until al dente. In the last 3 minutes of cooking, add the broccoli florets to the pasta water. Drain well and set aside.

While pasta cooks, season the chicken cubes with salt and pepper. Heat a large skillet over medium-high heat with a tablespoon of olive oil. Add the chicken and cook until golden brown and cooked through, about 5-7 minutes. Remove from skillet and set aside.

In the same skillet, melt the butter over medium heat. Add the minced garlic and cook until fragrant, about 30 seconds. Slowly whisk in the heavy cream and bring to a simmer. Cook for 3-4 minutes, stirring occasionally, until slightly thickened.

Remove the skillet from the heat. Gradually whisk in the grated Parmesan cheese until the sauce is smooth. Stir in the salt, pepper, and nutmeg.

In a large bowl, combine the cooked pasta and broccoli, the cooked chicken, and the Alfredo sauce. Stir gently to coat everything evenly. Pour the mixture into a 9x13-inch baking dish.

Sprinkle the shredded mozzarella cheese over the top. Bake for 20-25 minutes, or until the sauce is bubbly and the cheese is melted and lightly browned. Let it rest for 5-10 minutes before serving.