Step 1: Preheat oven to 350°F. Butter an 8 inch square or round pan, line bottom with parchment paper, and butter it. flour inside of pan, tapping out excess. In a medium bowl, whisk together 1 1/2 cups flour, the baking powder, ground ginger, baking soda, and salt.
Step 2: In a large bowl, whisk granulated sugar, 1/4 cup melted butter, the oil, carrot juice, eggs, and fresh ginger until combined. Add flour mixture and whisk just until smooth. Stir in carrots and scrape batter into pan.
Step 3: Bake until a toothpick inserted in center of cake comes out clean, 40 to 45 minutes. Let cake cool in pan on a rack 10 minutes. Run a paring knife around edges to loosen, then turn out onto rack. Turn right side up and let cool completely, at least 1 3/4 hours.
Step 4: In a small bowl, stir together powdered sugar, lime zest, and 1 tablespoon lime juice until smooth. Add more lime juice (about 1 teaspoon) for a thick, flowing texture. Drizzle over cake.
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