1) Preheat the oven to 375°F. Lightly grease a standard 12-well muffin pan or two muffin-top pans. Or line the pan with paper muffin cups, and grease the cups.
2) Whisk together the sugar, eggs, oil, and applesauce until well blended. Stir in the grated carrots and pineapple.
3) Whisk together the dry ingredients, including raisins and walnuts, in a large bowl.
4) Add the wet ingredients, stirring just to combine.
5) Divide the batter among the muffin cups, filling them quite full.
6) Bake the muffins until a toothpick inserted in the center of one comes out clean, about 22 minutes.
7) Remove the muffins from the oven, and cool in the pan for 5 minutes before transferring to a rack to cool completely.