Preheat the oven to 375 degrees F (190 degrees C).

Microwave milk in a small microwave-safe bowl until hot, about 3 minutes. Add hot milk to eggs, whisking until blended.

Place French toast cubes in a large bowl. Pour egg mixture over the cubes and toss gently.

Mix ham, turkey, Cheddar cheese, and Colby cheese together in a bowl. Layer 1/2 of the French toast cubes in the bottom of a 9x13-inch baking dish. Cover with 1/2 of the ham-cheese mixture. Repeat with remaining ingredients.

Bake in the preheated oven until top is golden brown and center is set, 30 to 35 minutes. Transfer to a cooling rack; cool for 10 minutes. Sprinkle powdered sugar over the top.

Combine raspberries and raspberry preserves in a small bowl. Cut casserole into square pieces; top each serving with raspberry mixture.