Preheat oven to 350 degrees F.
In electric mixer, combine butter with both sugars. Beat until creamed. Add egg and vanilla extract. Beat for 1-2 minutes.
Add flour, baking soda, cinnamon, ginger, and salt. Beat at low until combined. Stir in carrots and walnuts.
Scoop dough onto parchment-lined baking sheet and bake for 15-17 minutes (or until lightly golden).
Beat (medium speed) butter and cream cheese until fluffy. Add powdered sugar (1 cup at a time) until desired frosting consistency. For a looser frosting, use less powdered sugar. For a stiffer (for piping) frosting, add more powdered sugar. Once cookies have cooled, add frosting and garnish with walnuts.