egg-wash role in having soft bread buns

The egg wash is very important for the crust of your buns.

Different types of glazes and egg washes produce different results on bread's (or pastry's) crust, due to the combinations of proteins, fats, and sugars they contain. I would guess that one of two things is the problem with a milk glaze.

1. In your case, the baking temperature is too high for a milk wash. The sugars in the milk caramelize and the crust dries out and becomes hard before the inside of the bun is cooked.
2. You over baked your buns. An egg wash would darken in color sooner than a milk wash, so if you waited for the same depth of color as you had with the egg wash, you would have waited longer and the crust would have dried out more, becoming harder and crustier.

I think a combination of the two is most likely. A quick google will turn up a variety of articles about the different kinds of egg washes and glazes and their effect on bread, but I would start here and here.

If any further experiments also turn out results that are too crusty (or you decide to stick with a milk glaze), you should note that you can also brush the finished but still warm buns with a fat of some kind, like melted butter, which will soften the crust. This may salvage a crusty batch of buns.

I don't think that egg-wash is your only option, as brushing with milk should also work. After brushing with milk, you can follow the below steps to ensure a sift crust:

1. Brush with butter after baking

2. Wrap the loaf in plastic wrap while it cools to retain moisture

Just a note, a Pullman loaf pan might come in handy when making bread with a soft crust:

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