Preheat oven to 350 degrees F.
Combine the shredded carrots, applesauce, vegetable oil, and vanilla in large bowl. Although I almost always used my stand mixer for recipes, I mixed this in a regular mixing bowl because it's all very easily mixed by hand.
Add eggs and sugar to the wet ingredients and mix well.
In a different bowl, mix together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and walnuts. Mix these together until well blended.
Add the dry ingredients to the wet ingredients bowl.
Do not over mix them. Just combine them until the dry ingredients, well, until they no longer look dry.
Use liners with your muffin tins or grease the pan. I use my Silpat muffin pans, so I don't use liners or have to grease the pan. Bonus!
Fill each muffin well 1/2 - 3/4 full.
Bake muffins for 24 - 28 minutes. Mine take right around 27 minutes, but I would definitely keep an eye on yours until you know how long they take in your oven.
Allow to cool and then enjoy them!