1. Preheat oven to 350 degrees. Coat a 12 1/2-by-17 1/2-inch rimmed baking sheet with cooking spray. Line with parchment; spray parchment. Whisk together carrots, eggs, sugars, oil, and fresh ginger. In another bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, and ground ginger. Fold flour mixture into carrot mixture.

2. Spread batter in pan. Bake until golden and a toothpick comes out clean, about 20 minutes. Let cool in pan on a wire rack 20 minutes. Invert onto rack; let cool completely. Cut out 16 cake rounds to fit in an 8-to-10-ounce glass, piecing together scraps as necessary.

3. Spoon 2 tablespoons jam into each glass; layer with 1/4 cup filling, then 2 cake rounds, pressing to compact. Top each with 2 tablespoons filling and serve.