In a 3- to 4-quart pan over medium-high heat, stir oil, onion, carrots, and garlic until onion is limp, 5 to 7 minutes. Stir in white wine and bring to a boil. Pour mixture into a large bowl.
To pan, add broth and vermouth. Place over high heat, cover, and bring to a boil. Add squash. Cover and simmer over low heat until squash is tender when pierced, 10 to 12 minutes.
A portion at a time, whirl squash mixture in a blender until smooth. As purèed, add to onion mixture.
To serve hot, return soup to pan and stir over high heat until hot, 2 to 3 minutes. To serve cool, chill quickly by nesting pan in ice water; stir frequently until cold, 20 to 25 minutes. If making ahead, cover and chill up to 1 day. Add salt and pepper to taste.