In an 8-quart stockpot, add the butter and oil and melt together over medium-high heat. Add the onion and carrot and cook, stirring occasionally, until the onion is soft, about 5 minutes. Stir in the garlic and cook until aromatic, about 30 seconds. Add the squash and the chicken stock. Bring the mixture to a boil and add the sage. Continue to boil until the vegetables are tender, about 20 minutes. Turn off the heat. Using an immersion blender, blend the mixture until smooth and thick. Season with salt and pepper, to taste. Keep the soup warm over low heat.
For the crostini: Put an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Arrange the bread slices on a baking sheet. Drizzle with olive oil and sprinkle with sage. Sprinkle the cheese on top and season with salt, to taste. Bake until the cheese has melted and the bread is light golden, about 6 to 8 minutes.
To serve, ladle the soup into bowls and garnish with the cheese crostini.
Cook's Note: The cooked vegetable mixture can also be pureed (after cooling for about 5 minutes) by ladling, in batches, into a food processor or blender and blended until smooth.