Egg substitue in enriched breads I'm making Kolaches (a Czech semi-sweetened dough with a sweet filling), but I'm challenging myself to cook vegan, and therefore can't use an egg.  I've got a bunch of recipes, but all of them are semi-enriched, i.e. have egg, milk, and added fat.  I've got replacements for everything else, except the egg. 
What are things people have used for eggs in yeast breads.  It shouldn't need to be a binder or leaven, just enrich.  I've seen suggestions for sweet potato, just add oil, tofu.  Anyone have any experience with enriching yeast doughs without eggs?
I use ground flaxseed and water, which is a common egg substitute - I prefer the flavor of the "golden" or yellow seeds .vs. the brown ones (and unlike the brown seed grower's propaganda, I do taste a difference) but either will work for the egg substitution - chia seed is evidently another common substitute.

I consider it unsafe (or unwise) to assume that the only function of the eggs is to "enrich" - I find that the "same" bread made with or without eggs behaves quite differently in terms of how it rises, so I would suggest sticking to a substitute that gives you more complete functional replacement, rather than, say, "add some oil."

I'm not myself a vegan but I have baked things with keeping vegans happy in mind. As such I have worked with eggs and with ground flaxseed and water, and so far I consider the flaxseed mixture as good of an egg substitute as I have used, though still not the same as (nor indistinguishable from) actual eggs.