Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add cauliflower, cover, and steam until tender, about 5 minutes. Transfer cauliflower to a food processor; process using the shredding blade. Transfer cauliflower to the bottom of a 9x13-inch baking pan.

Preheat the oven to 350 degrees F (175 degrees C).

Refill steamer with water and return to a boil. Add broccoli, cover, and steam until tender, about 5 minutes. Arrange broccoli on top of the shredded cauliflower. Cover with cooked chicken.

Combine Alfredo sauce, 1 cup Cheddar cheese, onion soup mix, and garlic in a saucepan over medium heat. Cook and stir until hot and melted, about 2 minutes. Spread sauce over the chicken. Top with remaining Cheddar cheese.

Bake in the preheated oven until bubbly on top, 30 to 40 minutes.