Step 1

Preheat oven to 400 degrees F. Melt 1 tablespoon butter in a large skillet over medium-high heat. Add chicken to the pan; cook, stirring occasionally, until well browned, about 8 minutes. Transfer the chicken to a medium bowl. Add zucchini and bell pepper to the pan; cook, stirring occasionally, until the vegetables start to soften, about 4 minutes. Transfer the zucchini mixture to the bowl with the chicken.

Step 2

Add the remaining 2 tablespoons butter to the pan. Stir in flour; cook, stirring constantly, until the flour starts to brown, about 1 minute. Add broth and milk; bring to a boil, whisking often. Remove from heat and add cream cheese and 3/4 cup mozzarella; stir until melted. Stir in pepper and salt. Drain liquid from the chicken and vegetable mixture; stir the chicken and vegetables into the cheese sauce. Transfer to a 2-quart baking dish. Place the dish on a foil-lined baking sheet; sprinkle the casserole with the remaining 1/2 cup cheese.

Step 3

Bake until the top is browned and the edges are bubbly, 20 to 25 minutes. Let stand for 10 minutes before serving.