To make the stock, cut winter squash in half. Scrape seeds, and membrane out with a large metal spoon. Place what you've scraped out in a large soup pot. Add remaining stock ingredients (carrots through water) and bring to boil over medium-high heat. Reduce heat and simmer 25 minutes. Strain and set aside.
Meanwhile, preheat oven to 400 degrees °F.
Line a large baking dish with parchment paper. Place the squash halves, cut side down, in baking dish. Bake for 45 minutes to 1 hour, until skin is blistered and flesh is soft. When cool enough to handle, peel off and discard the skin.
Melt butter in soup pot. Add onion; cook and stir over medium heat until softened, about 5 minutes. Add cooked squash and stock and bring to boil. Reduce heat and simmer, covered for 20 minutes. If desired, puree some of all of the soup with a standard or immersion blender. Remove from heat and stir in heavy cream and cheese.
Season to taste with salt and freshly ground black pepper. Serve.