Bring eggs to room temperature. Peel the carrots and grate them on a grater with small holes. Peel the apple and grate it the same way as the carrots.
Preheat the oven to 180°C. Line an oblong pan 28-30 cm long with parchment paper.
Beat eggs with sugar until light and fluffy. While continuing to beat, pour in the oil in a thin, steady stream.
In another bowl, mix flour with baking powder, baking soda, cinnamon, ginger, and salt.
Pour into the beaten mixture and mix on low speed with the mixer, just until the ingredients combine.
Add carrots with the apple and mix with a spoon until you get a uniform batter. Pour into the pan and place in the oven for about 50 minutes or until a toothpick comes out clean.
Beat the cheese together with powdered sugar and vanilla extract for about 2 minutes, then spread over the cooled bread. Sprinkle with chopped nuts.
