Melt butter in a medium stockpot over medium-high heat. Add the leek and half the garlic. Season well with salt and pepper; cook, stirring until soft and tender, 3 to 5 minutes.
Add squash, and cook, stirring, 3 to 5 minutes. Add chicken stock, and bring to a boil. Reduce heat to a simmer, and cook until tender, 15 to 20 minutes. Remove soup from heat; let cool slightly.
Fill the bowl of a food processor with basil and remaining garlic. Drizzle olive oil through the feed tube; process until smooth. Season with salt and pepper. Transfer basil pistou to a small bowl.
Place slightly cooled soup in the cleaned bowl of the food processor, and process until soup is pureed but still slightly chunky.
Return to stockpot; cook to reduce liquid and thicken just slightly, about 5 minutes. Adjust seasoning with salt and pepper. Ladle into soup bowls, garnish with basil, and serve with basil pistou.