Preheat the oven to 375 degrees F (190 degrees C).
Combine 1/2 the mushroom soup and an equal amount of water in a saucepan over medium heat. Add chicken. Bring to a simmer and cook until chicken is no longer pink in the center, 3 to 5 minutes.
Heat 1 frozen, unopened bag of cauliflower and broccoli mix in the microwave until hot, 4 to 6 minutes. Let stand about 2 minutes; open bag and transfer vegetables to a bowl. Repeat with remaining bag.
Melt butter in a saucepan over medium heat. Add milk and salt; stir in remaining cream of mushroom soup. Add flour slowly, stirring constantly with a whisk to prevent lumps. Add Cheddar cheese. Cook and stir until cheese sauce is thick, 5 to 7 minutes. Add garlic powder.
Layer half the cooked vegetables into the bottom of a baking pan or casserole dish. Add half of the cooked chicken. Pour in half the cheese sauce. Add a light layer of bread crumbs. Add remaining vegetables, remaining chicken, and remaining cheese sauce. Top with a light layer of bread crumbs. Cover pan with aluminum foil.
Bake in the preheated oven until bubbling, 20 to 30 minutes.
The directions for steaming frozen vegetables in a microwave are based on the particular brand mentioned. Follow instructions on the bag if using a different brand. Cooking time will depend on the wattage of the microwave oven you are using.
Per Serving: 330 calories;18 g fat;
20.1 g carbohydrates;
20.2 g protein;
69 mg cholesterol;
641 mg sodium.