Heat oven to 400 degrees F. Grease baking sheet.
Place squash cut-side down on oiled baking sheet and bake until the squash can be pierced with a fork (about 45 minutes). Let cool, then scoop out the flesh into a soup pot, discarding the skin. Add vegetable broth and water and simmer uncovered, 30 to 40 minutes, stirring occasionally to break up the squash. Add salt and pepper to taste and, if too thick, add additional water or broth. Cool. Puree in food processor or hand-held blender until smooth, return to pot and heat. Serve drizzled with a ring of balsamic vinegar.