Good question @mfg. I hope I can help a little bit. The basics behind the egg wash are to provide for a couple different thing. These being shine, crispness, and color.
Shine
The shine is primarily provided by the egg yolk. The higher concentration of yolk the more shine.
Crispness
This come from the egg whites. The whites make things a little crispy and sometimes can crackle a little bit (mainly when usually by themselves).
Color
This comes from the fat and protein. Leaving the yoke in along with the shine will add color. You can add water to lighten the color a bit. I find that using the white also dilutes the color a bit but not as much (but the white also makes it crisp). Add cream or milk to get it a little bit darker.
Salt
I often find people adding salt to an egg wash. I does provide some flavor to the crust, but I have noticed that for some reason (when using a whole egg) it does allow it to get a little darker than it would normally.
Sugar
Like salt it adds flavor (sweetness) to the crust. And it will also add some color. Especially if baking at a higher temp and using a raw or brown type sugar.
Alcohol
I have seen the use of alcohol quite a number of times. From the results I have seen it behaves like water. Although not the the same degree. My guess is that it has to do with how fast the alcohol evaporates. As far as flavor, I can't say that it added much at the levels used. I did once definitely get a hint of a bourbon being used in the wash on a pretty plain white loaf. But they did mix in quite a good amount of bourbon to just an egg yolk.
That is all I have really used myself or seen. I am sure that you can use other liquids although I don't know what vinegar would do. Just keep in mind if they are high in fat, protein, or sugar they will add color. Otherwise they will lighten the color.
I have also seen melted butter just used as a wash. It adds a nice color and buttery flavor. I haven't seen it added to egg wash but I would assume you could add to egg yolk to get a shine and probably darker color than cream/milk.
Another important thing to remember is to beat your wash well when using for a bread coating. Unlike if your just using the was was to seal say a ravioli. If your egg was is spotty not one nice fully incorporated mixture your crust can also be spotty.
Doughs are docked to keep them from blowing up with steam while they bake.
Thus- you only do it in applications that you don't want blown up- like blind pie crusts. Puff pastry applications, for example, you usually do want to blow up so you will get a lot of light layers.
If you are baking a pastry with a filling then the filling will keep this from happening. If you are baking blind, even with pie weights, you should dock because the weights might not weight evenly and you might still get a bubble.
Best Answer
Milk contains several components, most notably fat, sugar, and protein. These all add to the browning effect when cooked at high temperature (proteins cooked together with either sugars or fats undergo the "Maillard reaction"). The browned bits do have significant extra flavor, but it does not taste like milk.
Egg white, by itself, is pure protein. It is typically used in pre-baked ("blind baked") pie shells because the dense protein creates a water-resistant coating when it cooks, so that the crust won't get too soggy when a liquid filling is added later on. Quiche is a great example of this application. Like milk, the egg white will add no flavor of its own, just the flavor associated with browning. Egg white is also used in low-cholesterol cooking.
Egg whites will brown some, but not as much as milk unless the pastry itself provides extra fats and sugars for the proteins to react with. Egg wash -- beaten whole eggs, usually mixed with a little water -- is more typically used to get pastries to brown.