Preheat the oven to 350 degrees F (175 degrees C). Line an 18-cup muffin tin with paper liners.
Mix eggs, applesauce, coconut oil, and honey together in a large bowl. Stir in brown sugar and white sugar. Mix in flour, baking powder, baking soda, nutmeg, and cinnamon slowly. Roll in carrots, walnuts, and raisins. Fill each muffin cup 3/4-full with batter.
Bake in the preheated oven until a toothpick inserted into a cupcake comes out clean, 25 to 30 minutes.