Preheat the oven to 350 degrees. Grease the cake pans with nonstick cooking spray and butter and evenly coat with flour, tapping out the excess.
Coarsely grate the carrots (you should have about 3 cups) and set aside.
Spread the walnuts in one layer in a small baking pan and put in the (preheated) oven. Set a timer for 10 minutes and check the nuts to see if they're a light golden brown. If not, toast 2 minutes longer. Let the nuts cool. Chop by hand into 1/4-inch pieces and set aside.
Over a bowl or piece of parchment, sift the flour, baking soda, cinnamon and ginger together in a sifter or in a fine strainer by gently tapping your hand against the edge. Add the salt (you can just leave it on top of the flour pile because it gets mixed in later). Set aside.
Using the stand mixer, or a handheld mixer and medium bowl, first on low speed and then gradually increasing the speed to medium, beat the butter, granulated sugar and brown sugar together until smooth. Scrape down the side of the bowl with the rubber spatula.
In a small bowl, whisk the eggs and vanilla together until well blended.
Alternating between the sifted dry ingredients and the eggs, add them to the butter/sugar mixture in thirds, making sure the batter is combined before the next addition.
With the rubber spatula, stir in the grated carrots and chopped walnuts.
Evenly divide the batter between the prepared pans and spread evenly with the spatula. Bake until a bamboo skewer inserted in the middle of the cakes comes out clean, about 40 minutes. Let cool completely.
To remove a cake from the pan, run a knife around the inside edge, then invert it and gently tap one side against the counter, letting the cake gently fall out of the pan onto a platter or piece of parchment paper. The cakes can be wrapped in plastic and kept at room temperature for 2 days or frozen for 2 months.
To assemble the cake, put one of the cake layers bottom-side up on the cake plate or platter. Using the narrow spatula, spread with about 1 cup of the frosting. Using the large spatula, place the other cake (also bottom-side up) on top of the frosting. Spread the frosting over the side and then the top of the cake.
Cream Cheese Frosting
1 3/4 cups confectioners' sugar
12 oz. cream cheese, softened
12 tbsp. (1 1/2 sticks) unsalted butter, softened
1 tsp. pure vanilla extract
Large pinch kosher salt
Sift the confectioners' sugar through a strainer to remove any lumps. Using the stand mixer, or handheld mixer and medium bowl, on medium speed whip the cream cheese and butter until smooth. Add the sugar, vanilla extract and salt and mix in thoroughly.