Preheat the oven to 500°F.
Arrange the tomatoes, peppers, and garlic cloves on a baking sheet. Roast until nicely deeply brown caramelized, 30 to 40 minutes. Remove from the oven and let cool for 10 to 15 minutes. Transfer the roasted vegetables* to a blender or food processor and pulse until it is a smooth paste.
In a large pot, heat the butter and olive oil over medium heat. Add the flour and cook for 1 to 2 minutes, stirring constantly with a wooden spoon or spatula. Add the tomato paste and continue to cook and stir for an additional 1 to 2 minutes. While whisking, add 1 cup of the chicken broth, until the flour mixture is evenly incorporated. Add the remaining 3 cups chicken broth, the water, and roasted vegetable paste. Season with salt, sugar, and black pepper. Bring to a boil and then add the cream and fresh basil. Add additional salt and sugar to taste.
To serve, ladle into bowls. Garnish with parmesan cheese and drizzle with olive oil, heavy cream, and fresh herbs as desired.