Fried carrot cake isn't a weird, deep-fried version of the American-style cake that is made with grated carrots and a cream cheese icing. Rather, it is a hawker-style dish served in Malaysia and Singapore. Its main ingredients are a steamed cake made of grated radish (known as "white carrot") and rice flour, which are stir-fried with bean sprouts, eggs and Chiu Chow-preserved radish.
Fried carrot cake is Chiu Chow in origin and it can be ordered either "black" (char kway) or "white" (chai tow kway). First make the steamed cake, which is similar to the loh bok goh (radish cake) served at dim sum restaurants, only it doesn't contain the dried meats and seafood. Let it cool then cut it into cubes. In addition to bean sprouts, eggs and pungent salted radish, both versions usually contain spring onion and garlic, but the white one is seasoned with fish sauce while the black one takes kecap manis (thick, dark soy sauce). Many versions also include shrimp.
If you buy fried carrot cake from hawkers, they'll scoop it onto a plate lined with grease paper, and you add a big dollop of chilli sauce.
In Hong Kong, my favourite version of this is served at Café Malacca, at the Hotel Jen, in Shek Tong Tsui, Western District.