Preheat the oven to 375°F (190°C).
In a large pot, bring salted water to a boil and cook the fettuccine according to package instructions until al dente. Add the broccoli florets during the last 3 minutes of cooking. Drain and set aside.
In a skillet over medium heat, melt the butter and sauté the minced garlic until fragrant (about 1 minute).
Stir in the heavy cream, salt, black pepper, and nutmeg. Cook for 2 minutes, allowing the sauce to thicken slightly.
Remove the skillet from heat and mix in the Parmesan cheese until melted and smooth.
In a mixing bowl, combine the cooked fettuccine, broccoli, shredded chicken, and the Alfredo sauce. Mix well to coat everything evenly.
Transfer the mixture into a greased baking dish. Sprinkle the mozzarella cheese evenly over the top, and if desired, add the breadcrumbs for an extra crunchy topping.
Bake in the preheated oven for 25-30 minutes, until the cheese is bubbly and golden.
Remove from the oven and allow to cool for a few minutes before serving.