Place the squash halves cut side up on a baking sheet and brush the insides with 1/2 tablespoon of the olive oil. Sprinkle evenly with 1/4 teaspoon salt, a few turns of black pepper, the cinnamon, paprika, cayenne, and cloves and bake until a fork pokes easily into the center. Begin checking for doneness after 1 hour.
Meanwhile, in a large pot, heat the remaining 2 tablespoons olive oil over medium heat. Add the onion and remaining 1/2 teaspoon salt. Cook, stirring, until the onion is soft and translucent, 5 to 7 minutes. Add the garlic and thyme and cook for 2 more minutes. Transfer the mixture to a blender or food processor, add the tomatoes with their juices, and blend until very smooth. Return the mixture to the pot.
Scoop the insides out of the squash and place it in the blender with the vegetable broth and blend until very smooth. Add it to the pot, bring it to a boil over high heat, then reduce to a simmer. Simmer, covered, for 20 minutes, stirring occasionally. Taste and adjust seasonings as desired. Ladle into bowls, drizzle with olive oil, and top with grated Parmesan.