Preheat oven to 420°. Line a 12-well muffin pan with paper liners.
Mix together applesauce, sugar, and oil. Add the eggs, one at a time, beating after each addition.
Sift together the dry ingredients, add them to the egg mixture, and mix well. Fold in the grated carrots. Fill the wells of the muffin pan to the top.
When you place the muffin pan in the oven, reduce the heat to 375°, and bake for approximately 18 minutes. Remove from the oven, and let sit for 5 minutes. Take the muffins out of the pan and allow to cool before serving.