This chilled soup is made with chayote, a squash native to Central America.
Melt butter in a stockpot over medium heat. Add leeks and onion; cook until translucent, about 2 minutes. Add garlic and jalapeno; cook 2 minutes. Add chayote, cilantro, and enough water to cover (about 8 cups); bring to a boil over medium-high heat. Reduce heat; simmer until squash is soft, about 25 minutes.
Prepare an ice bath. Puree mixture in a blender, working in batches so as not to fill more than halfway. Pass through a fine sieve into a large bowl set in ice bath. Season with salt; let cool completely, stirring frequently. Stir in cream.
Just before serving, core apples and cut into 1/4-inch dice; stir half into soup. Garnish servings with remaining apple.