I would like to cook finger food for my one year old. I am looking for a recipe for a soft cake-- something like a loose cheesecake or set custard in terms of consistency. I did make a bread pudding serving the middle portion to him, but I would like to avoid using so much bread or flour. I would adapt the recipe to accommodate savory vegetables or do a low sugar, or no sugar, option when using it for fruits. Anyone know the name of a bread/cake/tart that would come close to what I am after? 
You cannot simply leave out the sugar in a cake or pudding recipe, as it is quite important for the setting and the final texture. If you are looking for something which is made out of custard, has vegetables, and can be held in the fingers, you are describing a quiche. It can be made as mini-quiches too, for less messy eating.  