Preheat the oven to 350 F
In a large bowl, combine the grated carrots, sugar, oil, applesauce, eggs and vanilla extract. Whisk together until smooth
Add in the flour, cinnamon, baking soda, baking powder and salt and whisk together until fully incorporated
Fold in the walnuts, reserving some to top the muffins with.
Fill a 12-count muffin tin with 10 muffin liners and spoon about ¼ cup of batter into each cup. Top each muffin with the remaining chopped walnuts
Bake for 20-22 minutes, until the tops are firm to touch or a toothpick comes out clean
Let cool completely, then store in an airtight container for up to 3 days, or refrigerate for up to 1 week