For the carrot cake, peel carrots and grate finely with a kitchen grater. Preheat the oven to 180 degrees.
Separate eggs, beat egg whites until very stiff. Beat egg yolks with sugar, vanilla sugar and pinch of salt on highest speed for about 5 minutes until foamy. Stir in rum.
Mix flour with baking powder and half of the almonds and stir into the egg yolk-sugar mixture on lowest speed. Fold in beaten egg whites, then stir in remaining almonds and carrots as well.
Pour the batter into a loaf pan (23 x 10 cm, greased). Bake for 60 minutes. Let the finished cake rest in the pan for 10 minutes, then turn it out and let it cool.
For the icing, mix the powdered sugar and lemon juice to a thick paste, coat the cake and let it dry.