Set a 10-inch ovenproof skillet over medium heat and add 2 tablespoons of the butter. Once melted, add the broccoli and sauté until bright green and just tender, about 4 minutes. Remove the broccoli to a bowl.
Melt the remaining butter in the pan and add the garlic, cooking until fragrant, about 1 minute. Add the cream and bring to a simmer, then cook until reduced by about half, 3 to 5 minutes.
Stir in 1/2 cup of the mozzarella and all of the Parmesan until melted. Season with salt and pepper to taste. Stir in the broccoli and the chicken and top with the remaining 1 cup of mozzarella cheese.
Preheat the broiler. Set the pan about 6 inches from the heat and broil until the cheese is beginning to brown, 2 to 5 minutes (depends on how intense your broiler is!).