Preheat oven to 425 degrees Fahrenheit. Half the pumpkin and scope out all the seeds. (Keep them for roasting, if you'd like) Rub olive oil over the inside of the pumpkin and place cut sides down, onto a baking sheet. Roast for 35 minutes or longer, until the orange flesh is easily pierced through with a fork. Set aside to cool.
Heat 3 tablespoons olive oil in a large heavy-bottomed pot over medium heat. Once the oil is heated, add onion, garlic and salt. Stir occasionally, until onion is translucent, about 8 to 10 minutes.
Once cooled peel the pumpkin skin off the pumpkins and discard. Add the cinnamon, nutmeg, cloves, cayenne pepper and a bit of freshly ground black pepper to the pot. Lastly add the pumpkin flesh (or the pureed canned pumpkin). Pour in the broth. Bring the mixture to a boil, then reduce heat and simmer for about 20, allowing the flavors time to meld.
Once the mixture is done cooking, stir in the coconut milk and maple syrup. Remove the soup from heat and let it cool. Working in batches, transfer mixture to a blender (do not fill your blender past the maximum fill line or you will have pumpkin soup on your ceiling!). Securely fasten the blender's lid and use a towel to protect your hand from steam that escapes from the top of the blender as you purée the mixture until smooth.
Transfer puréed soup to a serving bowl and repeat with remaining batches. Taste and adjust if necessary (I added quite a bit more syrup, a tiny bit more salt and a splash more coconut. However this is all per preference of chef!)
Ladle soup into individual bowls and serve!