Preheat the oven to 425 degrees F. In a large ovenproof skillet over medium heat, add the garlic and about three-quarters of the scallions and cook, stirring occasionally, until soft, about 2 minutes. Add the broccoli, chicken, rice, tomato, 1 teaspoon salt, and pepper to taste; stir to combine.
Whisk the chicken broth and sour cream in a bowl, pour into the skillet and bring to a simmer. Stir in half each of the mozzerella and parmesan. Cover tightly with a lid or aluminum foil, transfer to the oven and bake until the rice is tender and most of the liquid is absorbed, about 20 minutes.
Turn on the broiler. Uncover the skillet and sprinkle with the remaining mozzerella and parmesan, then broil until golden, about 2 minutes. Sprinkle with the reserved scallions.