Heat olive oil in a 4- to 5-qt. pot over high heat. Add garlic, ginger, jalapeño, and salt. Cook, stirring constantly, until fragrant but not yet browned, 1 to 2 minutes. Add cayenne and cook, stirring, for 30 seconds. Add squash, broth, brown sugar, and 3 cups water. Bring to a boil, lower heat to a simmer, and cook, stirring occasionally, until squash is tender, about 20 minutes.
In a blender, purée the soup in batches until smooth, then pour back into the pot. Stir in cream and adjust seasonings to taste. Serve hot, with a swirl of crème fraîche if you like.
Note: Nutritional analysis is per serving.