Preheat oven to 400°F.
Melt 1 tablespoon butter in a large skillet over medium-high heat.
Add chicken pieces; cook, stirring occasionally, until well browned, 8 to 10 minutes.
Transfer the chicken to a medium bowl.
Add zucchini pieces, chopped bell pepper, minced garlic and 1 teaspoon Italian seasoning to the pan; cook, stirring occasionally, until the vegetables start to soften, 4 to 6 minutes.
Transfer the zucchini mixture to the bowl with the chicken.
Add the remaining 2 tablespoons butter to the pan.
Stir in ¼ cup flour; cook, stirring constantly, until the flour starts to brown, about 1 minute.
Add ¾ cup broth and ¾ cup milk; bring to a boil, whisking often.
Remove from heat and add 3 ounces cream cheese and ¾ cup mozzarella; stir until melted.
Stir in ¾ teaspoon pepper and ½ teaspoon plus ⅛ teaspoon salt.
Thoroughly 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 ½ cup mozzarella.
Bake until the top is browned and the edges are bubbly, 18 to 20 minutes.
Let stand for 10 minutes before serving.
