Preheat the oven to 350 degrees F. Line 12 muffin cups with paper liners.
Sift together the first 6 ingredients. In a large bowl, whisk the oil, brown sugar and eggs until well combined. Whisk in the applesauce, vanilla and carrots. Add the dry ingredients and mix until combined. Stir in 1/4 cup of the chopped walnuts.
Divide the batter between the muffin cups. Bake until a toothpick comes out clean, about 20 minutes. Transfer to a wire rack to cool completely.
With an electric mixer, beat together the cream cheese, confectioners' sugar and lemon zest until smooth and creamy. Frost the cooled cupcakes and sprinkle with the remaining 2 tablespoons chopped walnuts. The cupcakes should be stored in the refrigerator.