Preheat oven to 350°F. Place squash or pumpkin in a 9-by-13-inch baking dish. Trim top 1/2-inch from head of garlic (just enough to expose cloves) and add, cut side up, to baking dish. Drizzle squash and garlic with olive oil and sprinkle with salt and pepper. Cover dish with foil and bake 45 minutes. Remove foil and bake uncovered until garlic is golden and squash is very tender when pierced with a knife, 20 to 40 minutes longer.
Transfer squash to a food processor. When cool enough to handle, squeeze garlic from peels and add to squash. Pulse, adding broth 1/4 cup at a time, until desired consistency is reached. (You may not use all the broth.) Stir in butter and 1/4 tsp. each salt and pepper. Serve warm.