Peel squash and cut flesh into 1-in. cubes.
Heat oil in a large pot over medium heat. Add squash, onion, basil, and mint and cook, stirring occasionally, about 5 minutes. Add cinnamon stick, 1 tsp. salt, and 1 tbsp. ground chiles, followed by 4 cups stock and the spice sachet. Bring to a boil, lower heat to a simmer, and cook, partly covered, until squash is tender, 20 to 25 minutes. Remove spice sachet and cinnamon stick.
In batches, purée soup in a blender. Season to taste with salt.
Ladle soup into bowls. Swirl 1 tsp. or so of cream into each, leaving it streaky. Finish each with a pinch of ground chiles.
*Ground dried New Mexico chiles are fruitier and milder than other powdered chiles, such as cayenne. Try well-stocked grocery stores, or order Los Chileros mild ground chile (chile molido) from loschileros.com.
Make ahead: Through step 3, up to 2 days.