An "Asian cake" differs from any cake you can get at the grocery store in that it is much lighter in density and sweetness. The frosting is not overly sweet.
Diamond Bakery is my go-to for Asian cakes for birthdays and holidays. I've been going here for as long as I can remember!
Don't expect your cakes to be beautifully decorated, though. Even the frosting around the cake will be very imperfect. But it's fine if you don't look too closely. Service is iffy at times. Anyway, taste is good, and it's not super expensive (~$30 for an 8in cake). Other pastries are available, such as coconut cream buns and egg tarts. Come earlier in the day for these, as they'll run out!