Preheat oven 350 degrees. Line 12 muffin cups with paper liners. Sift together first 6 ingredients. In large bowl,whisk the oil, brown sugar and eggs until well combined. Whisk in the applesauce, vanilla and carrots. Add dry ingredents and mix until combined. Stir in 1/4 c. chopped walnuts. Divide batter into 12 muffin cups. Bake until toothpick comes out clean, approx. 20 min. Transfer to wire rack.
Frosting: With electric mixer, beat cream cheese, confectoners'' sugar and lemon zest until smooth and creamy. Frost cooled cupcakes and sprinkle with remaining 2 tbls. chopped walnuts. Store cupcakes in refrigerator.