Melt butter in a 6- to 8-quart wide heavy pot over moderate heat.

Cook onion with salt, stirring, until softened, about 8 minutes.

Add squash, carrots, potato, and chicken 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, puree soup in a blender until smooth.

Transfer pureed soup to a bowl.

Return pureed soup to the cleaned pot and thin with water if desired.

Simmer for 3 minutes.

Season with salt.

Pulse mint, parsley, and scallion in a food processor until finely chopped.

With motor running, add olive oil in a stream.

Add water and salt, blending until incorporated.

Swirl 1 tablespoon pistou into each bowl of soup.