Preheat oven to 400°F. Cook chicken in 1 tablespoon butter until browned, then transfer to a bowl.
Cook zucchini, bell pepper, and garlic in the pan with Italian seasoning until vegetables soften, then combine with chicken.
Make cheese sauce by cooking flour with remaining butter, then whisk in broth and milk; add cream cheese and mozzarella until melted. Stir in pepper and salt.
Drain excess liquid from chicken and vegetables, then mix into cheese sauce. Transfer to a baking dish and top with remaining mozzarella.
Bake for 18-20 minutes until browned and bubbly. Let stand 10 minutes before serving.
Notes
For a creamier texture, add a splash of extra milk before baking.