Preheat the oven to 375 degrees. Place the carrots in a microwave safe container with 1/2 cup water. Cook on high for 5-7 minutes or until soft. You can also lightly boil them with the pasta until soft.
Place the carrots with any remaining liquid in a blender. Add the 1 cup of cooking water from the pasta. Blend until smooth. Add the carrot puree to the pasta and cook over medium heat for 5 minutes. The sauce should thicken. Turn the heat down to low and stir in the 1.25 cups of cheese. Stir until creamy and season with salt and pepper.
Pour into a baking dish coated with cooking spray. Sprinkle with remaining cheese and breadcrumbs. Bake for 20-25 minutes until sides are bubbling. Broil for the last 2-3 minutes if you want a crispier top.