Cut off tops of squash (about 1 inch from stem end) and reserve. Scoop out seeds and discard. Cut a very thin slice off bottoms of squash to create a stable base. Brush bowls and tops all over with oil and sprinkle salt inside. Arrange squash bowls, with tops alongside, stem ends up, in 2 large shallow baking pans.
Roast squash in upper and lower thirds of oven, switching position of pans halfway through baking, until flesh of squash is just tender, about 1 1/4hours total.
Make soup while squash roast: Cook onion in oil in a 5-quart heavy saucepan over moderate heat, stirring occasionally, until softened. Add beets and apple and cook, stirring occasionally, 5 minutes. Add garlic and cook, stirring, 30 seconds.
Add broth and 4 cups water, then simmer, uncovered, until beets are tender, about 40 minutes. Stir in vinegar and brown sugar.
Purée soup in 3 batches in a blender until very smooth, at least 1 minute per batch (use caution when blending hot liquids), transferring to a large bowl. Return soup to pan, then season with salt and pepper and reheat. If soup is too thick, add enough water to thin to desired consistency.
Serve soup in squash bowls.
Cooks' notes: Squash flesh shrinks during baking; if a small hole forms, serve soup in squash but set in a soup bowl.
Soup can be made 3 days ahead and chilled, covered.