Preheat oven to 375°F. Coat a 12-cup muffin tin with cooking spray.
Combine 3 cups oats, 1 ¼ cups milk, ⅓ cup brown sugar, ¼ cup applesauce, 2 eggs, 1 tablespoon cinnamon, 1 teaspoon baking powder, 1 teaspoon vanilla, ½ teaspoon nutmeg and ½ teaspoon salt in a large bowl.
Fold in ½ cup carrot, ¼ cup raisins and ¼ cup walnuts.
Divide the batter between the prepared muffin cups, about ⅓ cup each. Bake until a toothpick inserted in the center comes out clean, 25 to 30 minutes. Run a knife around the edges of the cups to release the oatmeal cakes.
Cool in the pan for 10 minutes, then turn out onto a wire rack. Serve warm or at room temperature.