Puncture acorn squash and pumpkin in several places using a knife. Wrap the sweet potatoes in aluminum foil. Place squash, pumpkin, and sweet potatoes on a baking sheet.
Bake in the preheated oven until the pumpkin begins to cave in and the skin browns, about 2 hours.
Remove the skin, then chop the sweet potatoes. Skin, seed, and chop the pumpkin and acorn squash. Place the sweet potatoes, pumpkin, and acorn squash in a large pot and mash until smooth. Stir in the coconut milk until incorporated, reserving 1/2 cup for garnish, then stir in the chicken broth. Season with salt, pepper, lime juice, and ground ginger.
Cook over medium heat until heated through. Serve with a lime wedge and a drizzle of coconut milk.