Sift together the flours, baking soda, salt, cinnamon and nutmeg. In a large stand mixer with the paddle attachment, combine the brown sugar, oil and eggs until well combined. Add the carrots, applesauce and vanilla. Add the dry ingredients and mix until combined. Stir in 1/4 cup of the chopped walnuts.
Divide the batter among 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. Store any leftovers in an airtight container in the refrigerator.)