Bring a large pot of water to a rolling boil and salt it generously. Add the penne and cook for 6 minutes; stir in the broccoli florets and cook 2 minutes more, until the pasta is just shy of al dente and the broccoli is bright green and crisp-tender. Drain, reserving 1/4 cup of the pasta water. Heat the oven to 425°F.

While the pasta cooks, melt the butter in a medium saucepan over medium heat. Add the garlic and cook 1 minute, until fragrant but not browned. Pour in the evaporated milk and chicken broth and warm until steaming — do not let it boil. Take the pan off the heat.

Whisk the 3/4 cup Parmesan and the nutmeg into the warm milk until smooth and slightly thickened. Season with the salt and pepper. If the sauce looks too thick, whisk in a splash of the reserved pasta water.

In the baking dish, toss the drained pasta, broccoli, and shredded chicken with the alfredo sauce until everything is evenly coated. Spread into an even layer.

Shower the mozzarella and the remaining 2 tablespoons of Parmesan evenly over the top. Bake at 425°F for 8–10 minutes, until golden and bubbly around the edges. The chicken is already cooked — you are just heating everything through to steaming.

Rest 2–3 minutes so the sauce sets and thickens slightly. Scatter with parsley and cracked black pepper and serve straight from the dish.