Preheat oven to 375 degrees. Place liners in muffin cups and set aside. Shred the carrot and set aside.
In a bowl, whisk the eggs, one at a time. Add in the milk, applesauce and vanilla extract. Whisk until all ingredients are combined. Set the egg mixture aside.
In a large bowl, using a large spoon, combine the flour, sugar, baking powder, cinnamon and salt. Add the shredded carrots and mix well.
Melt the butter in the microwave for 20 seconds and stir. If more time is needed to melt the butter, do 10 second increments stirring after each one until the butter is melted.
Add the egg mixture and melted butter to the dry ingredients. Stir until ingredients are combined. Gently add in the chopped walnuts and raisins.
Spoon batter into prepared muffin cups. Fill to the top of each muffin cup.
Bake at 375 degrees for 20 minutes or until toothpick comes out clean.