Carrot Cake is more commonly call turnip cake in other Chinese communities.
It is a savory cake made with shredded diakon radish, rice flour, Chinese cured sausages, cured pork belly, dried shrimp and dried scallops. Some SG Chinese call them waxed sausage or waxed meat due to the term for cure sounds like the term for waxed. The cake is first steamed and can be eaten after the steaming. For whatever reason the freshly made cake is not popular in SG like other parts of SE Asia.
The SG versions are cut up and then stir fried. White version with egg and dark version is heavy on the dark soy sauce. Scallion and other "topping" added at the end. Chilli are added during the stir fry to maximize their flavours with the cake.