Heat oven to 400°F and cover a half baking sheet with aluminum foil.
Place the diced squash on the foil-covered baking sheet and drizzle with 1-2 tablespoons olive oil. Roast the squash 10 minutes, and remove from the oven. Set the squash aside until ready to use.
Meanwhile, in a heavy nonstick skillet, heat the oil over medium heat and add the onion, carrots and sweet red pepper. Cook (stirring often) until vegetables are tender and begin to turn golden brown, approximately 8 minutes. Add the garlic, stir and cook another 30 seconds. Add the chili powder and ground cumin and cook another 2-3 minutes, until mixture begins to stick to the pan, stirring often.
Add tomatoes and dried oregano and salt (to taste). Bring mixture to a boil, reduce heat to simmer and cook down just until mixture begins to stick to the pan, about 10 minutes. Stir often.
Stir in tomato paste & water mixture and bring the mixture back to a low boil. Season, to taste, with salt, and simmer, stirring often, 10 minutes or until mixture is fragrant and thickens.
Add the drained canned beans and the roasted squash. Bring to a boil. Keep temperature to a simmer over low heat, stirring often, for 30-45 minutes. The chili should NOT stick to the bottom of the pot. It will be thick, but if you prefer a thinner chili, add more water, stir and simmer to blend the flavors. Salt, to taste.
Add the chopped fresh cilantro 5 minutes before turning off the heat. Simmer for 5 minutes.
Spoon into individual bowls and top with shredded cheese.