1. Preheat oven to 350°.
2. To prepare cake, weigh or lightly spoon flours into dry measuring cups; level with a knife. Combine flours, xanthan gum, and next 6 ingredients (through nutmeg) in a medium bowl; stir with a whisk. Place granulated and brown sugars, oil, applesauce, 1/2 teaspoon vanilla, and eggs in a large bowl; beat with a mixer at medium speed until blended. Gradually add flour mixture, beating at low speed until smooth. Fold in carrot, raisins, and pecans.
3. Pour batter into an 11 x 7-inch metal baking pan coated with cooking spray. Bake at 350° for 35 minutes or until a wooden pick comes out clean. Cool completely in pan on a wire rack.
4. To prepare frosting, combine butter, 1 teaspoon vanilla, and cream cheese in a medium bowl; beat with a mixer at medium speed until blended. Gradually add powdered sugar, beating until smooth. Spread frosting over top of cake. Store cake, loosely covered, in refrigerator.