Make the cake: Preheat oven to 350 degrees. Prepare two 9" cake pans with non-stick spray. Whisk flour, baking soda, salt, cinnamon, ginger and nutmeg together in medium bowl. Set aside.
In bowl of stand mixer fitted with paddle attachment, combine sugar and oil until well blended. Add eggs, one at a time, to sugar-oil mixture, scraping bowl as needed between additions. Add flour mixture. Paddle until well combined. Remove bowl from mixer and fold in grated carrots until evenly combined.
Divide batter evenly between the two prepared cake pans. Bake until center is set and cake tester emerges cleanly from center, about 40 minutes. Cool cakes for 15 minutes in pans, then turn out onto a cooling rack. Allow cakes to cool completely before frosting.
Make the frosting: Combine cream cheese and butter in bowl of stand mixer fitted with paddle attachment. Paddle until very smooth. Add confectioner's sugar and vanilla. Paddle until combined, scraping down sides as needed. Paddle on high until aerated and fluffy, about 5 minutes.
Assemble the cake: Place flat side of one cake layer on serving dish. Spread a nice layer of cream cheese frosting on top of layer. Place second cake layer on top of frosting. Cover cake with remaining frosting.