1. Position a rack in the centre of the oven and preheat to 175°C (350°F). Grease a 23 x 33 cm (9 x 13-inch) baking pan and line the bottom with parchment paper.

2. In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt.

3. In a large bowl, whisk the eggs, brown sugar, and granulated sugar for about 1 minute. Gradually whisk in the oil, followed by the applesauce and vanilla.

4. Add the dry ingredients and fold with a spatula just until no dry flour remains. Gently fold in the grated carrots, taking care not to overmix.

5. Spread the batter evenly in the prepared pan. Bake for 35 to 40 minutes, until a toothpick inserted in the centre comes out with a few moist crumbs but no wet batter. Cool completely in the pan on a wire rack, about 2 hours.

6. For the icing, beat the butter for 2 minutes, until pale and creamy. Add the cream cheese and beat briefly until smooth. Gradually mix in the icing sugar, followed by the vanilla and salt. Avoid overbeating so the icing stays firm.

7. Spread the icing over the cooled cake. Refrigerate for 30 minutes before slicing for cleaner portions. Keep the cake covered in the refrigerator and let it stand at room temperature for about 20 minutes before serving.