Place a rack in the middle of the oven and preheat it to 180 °C / 350 °F. Grease a baking dish (20cm x 24cm or 8-inch x 9.5-inch) with butter and sprinkle with flour. Discard any excess flour.
First, make the cake. Grate the carrot into a large bowl. In a bowl, stir to combine sugars, flour, cinnamon powder, nutmeg powder, baking soda, and baking powder. In a separate bowl, combine eggs, both oils and apple sauce. While continuously whisking with a whisk, add the dry ingredients until the mixture is smooth and combined. Stir in the grated carrot and chopped walnuts.
Pour the carrot cake mixture into the prepared baking dish and spread evenly. Place in the oven. Bake for 45 minutes at 180 °C / 350 °F. When the cake is baked, remove it from the oven, and bring it to room temperature.
Mix powdered sugar, melted butter, and rum in a bowl. Mix for 5 minutes, then add in the cream cheese. Continue to mix for 3 - 4 minutes to get a smooth cream. Spread the cream over the cooled carrot cake. Grate the orange zest on top. Cut into slices and serve.