First, make the cakes:

1. Preheat the oven to 200C/400F. In a large bowl mix together the sugar, flour, baking powder salt and nutmeg. 
2. In another bowl, whisk together the egg and butter until well combined. 
3. Whisk the egg and butter mix into the flour mixture until you have a thick, even consistency. 
4. Grease the pudding tins with some butter and then spoon the mixture in. Place in the oven for 20-25 minutes. They are done when you can insert a skewer into the middle and it comes out clean. 
5. Leave to cool before serving.

While the cakes are baking, make the cream cheese frosting:

1. In a large bowl, whisk the cream cheese, butter and sugar together. 
2. Spoon into a piping bag and pipe over the cakes just before serving. The frosting can be stored in the fridge until ready to use

Either before or after making the cakes (depending on when you plan to eat them), make the candied carrot swirls:

1. Preheat the oven to 100C / 225F. Using a speed peeler, peel the carrot into thin strips. 
2. Slice down the long edges to leave a 1 cm thin strip from the middle of each piece.
3. Bring the sugar and water to a boil in a small saucepan. Add the carrot strips and simmer for 15 minutes. 
4. Tip the carrots into a fine-mesh sieve. Space the pieces out so they are flat and don’t stick together and leave to cool for 5 minutes. 
5. Line a baking tray with some baking paper. Spread the carrot strips out evenly and place in the oven for 30 minutes. 
6. Remove from the oven and wrap the pieces around a thick chopstick, or another circular stick, such as the end of a spoon. Gently pull off the stick, sprinkle with some sugar and leave the pieces to dry for 30 minutes. 
7. If the carrot pieces are still too soft, place them back in the oven for 10-minute intervals, until crisp.