For the cake: Adjust oven rack to middle position; heat oven to 350 degrees. Spray 13 by 9-inch baking pan (or 2, 9" round cake pans) with nonstick cooking spray. Line bottom of pan with parchment and spray parchment.
2. Whisk together flour, baking powder, baking soda, cinnamon, nutmeg, cloves, and salt in large bowl; set aside.
3. In food processor fitted with large shredding disk, shred carrots; transfer carrots to bowl and set aside. Wipe out food processor work bowl and fit with metal blade. Process granulated and brown sugars and eggs until frothy and thoroughly combined, about 20 seconds. With machine running, add oil through feed tube in steady stream. Process until mixture is light in color and well emulsified, about 20 seconds longer. Scrape mixture into medium bowl. Stir in carrots and dry ingredients until incorporated and no streaks of flour remain. Pour into prepared pan and bake until toothpick or skewer inserted into center of cake comes out clean, 35 to 40 minutes, rotating pan halfway through baking time. Cool cake to room temperature in pan on wire rack, about 2 hours.
4. For the frosting: When cake is cool, process cream cheese, butter, sour cream, and vanilla in clean food processor work-bowl until combined, about 5 seconds, scraping down bowl with rubber spatula as needed. Add confectioners' sugar and process until smooth, about 10 seconds.
5. Run paring knife around edge of cake to loosen from pan. Invert cake onto wire rack, peel off parchment, then invert again onto serving platter. Using icing spatula, spread frosting evenly over surface of cake. Cut into squares and serve. (Cover leftovers and refrigerate for up to 3 days.).