Preheat the oven to 350°F. Line 12 muffin cups with paper liners.
In a medium bowl, sift together both flours, the baking soda, salt, and spices. In a large bowl, whisk together the oil, brown sugar, and eggs until well combined. Whisk in the applesauce, vanilla, and carrots. Add the dry ingredients and mix until just combined. Stir in the walnuts.
Divide the batter between the muffin cups. Bake until a toothpick inserted in a cupcake comes out clean, about 20 minutes. Transfer to a wire rack to cool completely.
To make the frosting, 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 walnuts.
The cupcakes should be stored in the refrigerator, where they will keep for about 3 days.