Preheat the oven to 375 degrees . Grease three 2-by-8-inch round cake pans; line with parchment and grease the paper. Bring 4 cups water to a boil in a medium saucepan. Meanwhile, cut 1 lb. carrots crosswise 1/3-inch-thick. Using a box grater, shred the remaining 1/2 lb. carrots.
Add the sliced carrots to the saucepan. Cook until tender, 10 to 12 minutes. Drain and rinse with cold water until cool. Using a food processor, puree until almost smooth. In a large bowl,m whisk the flour, ginger, baking powder, baking soda, cinnamon and 1/2 tsp. salt.
Using an electric mixer, beat the butter at high sped until fluffy, about 2 minutes. Lower the speed to medium and gradually add 2 1/4 cups sugar; beat at high speed until fluffy, about 5 minutes. Beat in the whole eggs 1 at a time. Mix in 1 tsp. vanilla. At low speed, add half of the flour mixture, mixing until barely combined. Mix in the carrot puree and the remaining flour mixture. Fold in the shredded carrot.
Divide the batter among the pans; smooth the tops. Bake until springy in the center, about 25 minutes. let the layers cool in the pans for 10 minutes, then turn out onto racks and let cool completely before removing the parchment.
Make the frosting:
In a large metal bowl, combine the egg whites, cream of tartar and the remaining 1 1/2 cups sugar and 1/4 tsp. salt; stir in 1/2 cup cold water. Set the bowl over a large saucepan filled with simmering water and beat with an electric mixer at high speed until stiff, 5 to 7 minutes. Remove the bowl from the heat; beat in the remaining 2 tsp. vanilla. Continue beating until the frosting cools, about 3 minutes.
Place a cake layer right side up on a platter. Spread with a generous cup of frosting. Repeat with the remaining layers, frosting the top and sides of the cake with the remaining frosting. To serve, cut with a serrated knife, using a sawing motion.