Cut top off kabocha squash like a lid; reserve. Scoop out seeds and discard. Carve all but 1 inch of the flesh out of the squash.
Microwave hollowed kabocha on medium power in 5-minute intervals until tender but not too soft, about 15 minutes.
Heat 1 tablespoon oil in a saucepan. Add squash flesh; season with salt and pepper. Pour in enough water to cover and cook, uncovered, until squash is soft, about 15 minutes. Drain off most of the water. Puree with an immersion blender until creamy.
Preheat oven to 325 degrees F (165 degrees C). Line a rimmed baking pan with aluminum foil.
Heat remaining 1 tablespoon oil in a skillet. Add onion and garlic; cook and stir until softened, about 5 minutes. Add shrimp; season with salt and cayenne pepper; cook until shrimp turns pink, 5 to 6 minutes. Fold in pureed squash.
Spread most of the cream cheese inside the hollowed squash with a spoon. Ladle shrimp mixture in carefully. Top with remaining cream cheese. Cover with reserved top.
Bake in the preheated oven until heated through, 20 to 25 minutes.