When to apply a white egg wash on white wheat bread?
You are trying to counteract the formation of a crust - which is the opposite of what a good hot bake works towards.

The key to this is to keep the surface of the dough as moist as possible and this starts already after the shaping during the second rise. Apply the first wash at this point. I personally go for a whole egg, mixed with a splash or two of milk. This will reduce the “skin” that tends to form on the surface. Repeat the egg-washing right before you bake your bread. A generous steaming will also slow down the crust formation. 

If you don’t like the slight color caused by the egg yolks, using more milk or pure cream could be a start in the right direction.

But without eggs, it’s difficult to get a soft crust and that lacquer-like gloss. This is an enriched dough, twice brushed with cream and baked at 200 - 180 C for about 45 minutes:



The crust is super soft, though.

For hard crusts on bread, a quick spray or wipe down with water just a minute or two before you pull it out will be the simplest way.