Preheat oven to 350 degrees F (175 degrees C). Lightly grease a baking dish.
Place the spaghetti squash cut side down on a lightly greased baking sheet. Place in preheated oven, and bake until soft when a fork is inserted., 30 to 40 minutes.
Scoop out the cooked squash, and place in a mixing bowl. Add butter to the squash and stir until butter melts. Stir the mayonnaise, 1 cup Parmesan cheese, and Monterey Jack cheese into the squash mixture until well blended. Pour into the prepared baking dish. Sprinkle with 3 tablespoons Parmesan cheese.
Bake in preheated oven until cheese melts, about 20 minutes.