Make the cake: Position oven rack in the center of oven; preheat oven to 350°; butter the bottom and sides of two 9-inch round cake pans; dust the pans with flour.
Sift together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and cloves in to a bowl.
Whisk together the dry ingredients until blended; set aside.
In the bowl of an electric mixer, using the paddle attachment, beat the eggs, sugar, and light brown sugar at medium speed until well combined, about 2 minutes.
On low speed, add the oil, melted butter, milk, vanilla extract, ginger, and orange zest and mix until blended, scraping down the sides of the bowl with a rubber spatula as necessary.
Add the flour mixture in three additions, mixing just until blended.
Add the carrots and walnuts and mix until blended.
Scrape the batter into the prepared pans, dividing it evenly.
Bake the cakes for 25-30 minutes, until they are dark golden brown and a cake tester inserted into the center comes out clean.
Cool the cakes in the pans on wire racks for 15 minutes.
Invert the cakes onto the racks and cool completely.
Make the frosting: melt the white chocolate in the top of a double boiler over barely simmering water, stirring frequently.
Remove the chocolate from the heat and set aside to cool slightly.
In the bowl of an electric mixer, using the paddle attachment, beat the cream cheese and butter at medium speed until smooth and creamy, about 1 minute.
Add the melted white chocolate and mix until blended.
Add the lemon juice, vanilla, and powdered sugar; mix until blended.
Increase the speed to high and beat until smooth and creamy, about 1 minute.
Use the frosting immediately, as it will begin to harden quickly.
Frost the cake: place one of the cake layers right side up on a serving plate or cardboard cake round; spread the top of the cake with about 1 cup of frosting; top with the other cake layer, upside down.
Frost the top and sides of the cake with the remaining frosting.
Serve the cake immediately, or refrigerate; bring to room temperature before serving.