Make the cake: Preheat oven to 350 degrees. Butter three 9-inch round cake pans. Line bottoms with parchment paper, and butter parchment. Dust with flour, tapping out excess. Whisk together flour, baking powder, baking soda, cinnamon, salt, ginger, and nutmeg.
Beat butter and sugars with a mixer on medium speed until pale and fluffy. Add eggs, one at a time, beating well after each addition. Beat 3 minutes. Add vanilla, water, and carrots. Beat until well combined, about 2 minutes. Reduce speed to low, and add flour mixture, then finely chopped pecans.
Scrape batter into prepared pans, dividing evenly. Bake, rotating pans halfway through, until golden brown and a toothpick inserted into centers comes out clean, about 30 minutes. Let cool in pans on a wire rack for 15 minutes. Run a knife around edges of cakes to loosen, and turn out cakes onto rack. Turn right side up, and let cool completely.
Using a serrated knife, trim rounded top of 2 cakes. Place one trimmed cake, cut side up, on a serving platter. Spread 1 cup frosting over cake. Top with second trimmed cake, cut side down. Spread 1 cup frosting over cake. Top with remaining cake. Spread remaining frosting over top and sides. Gently press coarsely chopped pecans onto sides of cake. Refrigerate 1 hour before serving.
Refrigerate up to 1 day, covered.