Are there desserts which use whole eggs (not mixing them in)? It is my opinion that all savory dishes can be improved by putting a fried egg on top, and I've yet to find an exception.

However, the idea of doing that to a sweet dish seems totally foul. 

Considering that eggs are ingredients in many desserts, like cake batters, curds, and meringues, why would that be? It's not just that they're not sweetened, because you can use, for example, unsweetened cream cheese in both a sweet pastry or a savory one.

Am I wrong in my premise? Are there sweet dishes with eggs that are used without mixing them into the other ingredients?
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.