Preheat the oven to 375 degrees . Halve and seed the squash. Place cut side down on a foil-lined baking sheet and bake until soft, about 30 minutes. Let cool slightly, then scoop out the flesh and measure out 2 cups.
In a large pot, heat the EVOO over medium heat. Add the onion, 1 tbsp. basil and the dried mint. Cook, stirring occasionally, until the onion is soft, about 5 minutes. Stir in the squash, cinnamon stick, ground chile and 1 tsp. coarse salt. Stir in the stock or water and bring to a boil. Lower the heat to medium-low, cover and simmer, stirring occasionally, 25 minutes. Discard the cinnamon stick. Break up any large squash pieces with a spoon, or pulse in a blender or with an immersion blender to smooth.
Ladle the soup into bowls. Top with sliced fresh basil and mint, a pinch of chile and a drizzle of heavy cream.