Are there desserts which use whole eggs (not mixing them in)?
It's a cake rather than a dessert (always a tricky distinction) and traditionally uses the unbroken yolk rather than the whole egg, at but I offer you the Chinese moon cake. The egg isn't universal; neither is the cake being sweet. You certainly get sweet ones with egg. 

Apart from being a (rather poor) example, the Mooncake hints at something else: the sweet/savoury-main course /dessert distinction isn't universal. In fact it has European roots but has spread widely. 

Eggs also have small but non-negligible amounts of salt and glutamate, flavours we associate with savoury foods, especially together.