Place egg, 55g butter, flour, caster sugar, orange zest, and grated carrots into a food processor.
Blend ingredients briefly until just combined without overmixing.
Grease a 600ml pudding basin with butter.
Spoon mixture into the basin and smooth the surface with a wet palette knife.
Cover basin with cling film and microwave on high for 6 to 7 minutes.
Allow cake to cool before loosening edges and turning out onto a plate.
Combine yoghurt, icing sugar, and orange zest in a bowl.
Drizzle icing over the cake and garnish with mint.