Make soup: Melt butter in a 6- to 8-quart wide heavy pot over moderate heat, then cook onion with salt, stirring, until softened, about 8 minutes. Add squash, carrots, potato, and stock and bring to a boil. Reduce heat, then simmer, partially covered, until vegetables are very tender, about 20 minutes. Remove from heat and cool soup, uncovered, 10 minutes.
Working in batches, purée; soup in a blender until smooth (use caution when blending hot liquids) and transfer to a bowl. Return puré to cleaned pot and thin with water if desired; simmer 3 minutes. Season with salt.
Make pistou while vegetables simmer: Pulse mint, parsley, and scallion in a food processor until finely chopped. With motor running, add oil in a stream, then add water and salt, blending until incorporated.
Swirl 1 tablespoon pistou into each bowl of soup.
Cooks' notes: Soup (without pistou) can be made 2 days ahead and cooled, uncovered, then chilled, covered. Reheat before serving, thinning with additional water if desired. Pistou keeps, its surface covered with plastic wrap, chilled, 3 days. Bring to room temperature before using.