Preheat the oven to 425 degrees F. Cut the squash in half lengthwise and scoop out the seeds. Rub all over with 1 teaspoon olive oil and season the cut sides with salt and pepper. Arrange cut-side down on a baking sheet and roast until very tender, about 30 minutes (a fork should easily pierce the skin). Set aside until cool enough to handle, then scoop out the squash into a bowl and discard the skin.
Heat the remaining 2 tablespoons olive oil in a large pot or Dutch oven over medium-high heat. Add the tortilla pieces and cook, stirring occasionally, until browned, 3 to 4 minutes. Add the carrots, red onion, chile, garlic, cumin, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, until the onion has softened, about 4 minutes.
Add the roasted squash, broth and 3 cups water. Bring to a boil, reduce the heat to medium low and simmer until the carrots are very tender, about 25 minutes. Remove from the heat and let cool slightly.
Working in batches, transfer the soup to a blender and puree until smooth; return to the pot. (Or puree the soup in the pot with an immersion blender.) Add up to 1 cup water if the soup is too thick and reheat if needed. Season with salt and pepper. Serve with toppings.