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

Bake squash in the preheated oven until easily pierced with a knife, about 1 hour and 15 minutes. Remove from oven and let cool until easily handled. Scrape insides of squash into spaghetti strands with a fork and place into a large bowl.

Set an oven rack about 6 inches from the heat source and preheat the oven's broiler. Place chicken on a broiler pan sprayed with non-stick cooking spray (spray only where chicken will be placed). Brush chicken with 1 tablespoon melted butter and season with salt and pepper.

Broil on high for 4 minutes. Remove broil pan from oven, turn chicken over, brush with remaining 1 tablespoon butter and season with salt and pepper. Return to broiler for about 3 more minutes. Remove from oven and allow to cool. Set the oven to 350 degrees F (175 degrees C).

Pour condensed cream of chicken soup and chicken noodle soup into the bowl with the spaghetti squash strands. Add mozzarella cheese and Colby-Jack cheese and mix everything to combine. Dice cooked chicken and fold in.

Spray a 9x13-inch baking dish with cooking spray and spoon in spaghetti squash mixture.

Bake in the preheated oven until lightly browned and cheese is melted, about 45 minutes.