Preheat oven to 350 degrees. In a large bowl add eggs, oil, buttermilk, sugar, vanilla, orange juice, and lemon juice. Beat together, then mix in flour, baking soda, baking powder, and spices. Fold in shredded carrots, grated ginger, and nuts. Spread into two greased nine-inch pans*. Bake for 25 to 35 minutes. Cool in pans for 10 minutes then turn out onto wire racks. Allow cakes to cool before frosting.
Use a mandolin (or a vegetable peeler or sharp chef's knife) to slice thin pieces of ginger (with the bark removed). Place in a saucepan and simmer in 1/4 cup of water for 30 minutes or until ginger is tender. Remove from heat and drain water. Return ginger to pan, add 1/4 cup water and 1/4 cup of sugar and bring to a boil, stirring frequently. Reduce heat to low and continue to stir until the syrup begins to look dry (almost as if it has evaporated). Remove from heat and allow ginger pieces to cool on a wire baking rack.
You will need 15 ounces (421g) of grated carrots so buy more than 1 pound to allow for peeling and trimming end pieces.
*I found that the cake stuck on the bottom. Next time I'll line my cake pans with parchment paper and spritz with non-stick cooking spray before adding batter. If you have non-stick pans you probably won't need the parchment paper.
To easily grate fresh ginger, peel the outer layer off with a vegetable peeler then freeze the ginger root in a freezer-safe ziplock bag. Once frozen, remove as needed and grate on a rasp-style grater. The leftover ginger can be stored in the freezer to use for future recipes.
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