Cook, stirring occasionally, until onion is translucent, about 8 to 10 minutes. In the meantime, peel the pumpkin skin off the pumpkins and discard. Add the cinnamon, nutmeg, cloves, cayenne pepper and a few twists of freshly ground black pepper to the pot, followed by the pumpkin flesh. Use your stirring spoon to break up the pumpkin a bit. Pour in the broth. Bring the mixture to a boil, then reduce heat and simmer for about 15, to give the flavors time to meld.
Heat 3 tablespoons olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Once the oil is shimmering, add onion, garlic and salt to the skillet. Stir to combine.
Slice each pumpkin halve in half to make quarters. Brush or rub olive oil over the flesh of the pumpkin and place the quarters, cut sides down, onto the baking sheet. Roast for 35 minutes or longer, until the orange flesh is easily pierced through with a fork. Set squash aside to cool slightly.
Preheat oven to 425 degrees Fahrenheit and line a baking sheet with parchment paper for easy cleanup. Carefully halve the squash and scoop out the seeds, reserving them for roasting if you’d like (see note—you won’t need the roasted pumpkin seeds for this recipe).