Preheat the oven to 350°F and line 12 muffin cups with paper liners or lightly greased.
In a large bowl, stir together carrots, sugar, oil, applesauce, eggs and vanilla extract. Whisk together until smooth.
Add flour, cinnamon, baking soda, baking powder and salt and stir together until fully incorporated. Fold in the most of the walnuts, reserving some to top the muffins with.
Spoon equal amounts of batter into each cup and sprinkle with remaining walnuts.
Bake for 25 minutes or 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.