Peel the pumpkin or squash and cut into 1-inch pieces. (To make it easier to peel the pumpkin, cut in half, remove the seeds — and set aside for roasting! — and then microwave each half cut-side down in a shallow dish of water for 5 minutes, or roast the halves at 450°F for about 15 minutes. When the pumpkin is cool enough to handle, shave off the skin with a sharp knife.)

In a 6-quart or larger Dutch oven or soup pot, heat the butter and olive oil over medium heat. When melted, whisk in the cornmeal until smooth, then stir in the pumpkin, turnip, bell peppers, onion, garlic, and tomato paste. Cook, stirring frequently, for about 10 minutes.

Add the broth, diced tomatoes and their juices, beans, and corn. Stir in the chili powder, cinnamon, cumin, and Worcestershire sauce. Bring back to a simmer. Reduce the heat and simmer for at least 1 hour, or until the pumpkin and turnip are tender. (It gets even better if simmered longer — a whole afternoon, if you have the time.)

Before serving, taste and season with salt and pepper and a dash of balsamic vinegar. Serve hot with green onions, shredded cheddar cheese, and sour cream if desired.