Egg wash is a beaten egg that is diluted with milk or water. It can be lightly brushed onto pastries and breads before baking and provides a crispy, golden sheen to the bread. It can also be used to bind breading to deep-fried foods. There are many reasons a person may not be able to use eggs in their baking, from food allergies to diet. At such times, you may need an egg wash substitute to recreate the crunchy, tempting crust that an egg wash can provide.
Egg Wash Substitute
Probably, the best egg wash substitute is milk. You can also use custard and water, melted butter, olive oil, honey, maple syrup, yogurt, and vegan options like soy or almond milk.
Milk or Cream
One of the easiest egg wash substitutes is to simply brush on milk or cream on your dough before baking. This may recreate the color of an egg, without adding unwanted flavor or oil. Milk is often the preferred top layer for browning baked goods like scones. You can also add a dash of honey or agave syrup to the ‘milk wash’ for a sweet finish. 
To use: There are no rules here. Just apply a layer or two on the surface of your baked item with a pastry brush.
This is a popular egg wash substitute. There are few things that can beat the taste of butter. It gives a crispy delicious flavor to the finished dish.
To use: Melt butter at a low flame and turn it off just as the butter melts. Take care that you do not brown or burn the butter. Use the melted butter sparingly, as the butter will soak into the dough and can end up making it oily.
Almond milk, an alternative to traditional milk can also be used as a wash. This is probably one of the best options for vegans. It may give the pastry a dark gold hue, but also add a touch of nuttiness to the dough.
To use: Use as much as you require. You can apply a layer or two with a pastry brush.
Oil is another vegan substitute that may help in bringing luster to baked products. A light brushing of oil will give your dough a crunchy crust. A popular choice in this category is olive oil. Apart from a good shine, you can also get a crunchy texture. Coconut oil can also be used, especially for cookies. However, both of these have a distinct flavor. Pick a flavorless oil like canola oil if you don’t want any additional flavor. 
To use: Use lightly, to avoid over-greasing the dough or burning the crust.
Honey or Maple Syrup
Sweet options like honey or maple syrup may turn your pastry a deep gold, with lots of caramelized crunch. The sugar in the honey or maple syrup will caramelize, giving the finished product a lovely deep golden glow. Avoid using savory goods.
To use: These can burn very quickly in a hot oven. So it’s best used with quick recipes with short baking time. Alternatively, use them towards the end of the baking time.
Mayonnaise is a thick spread that works as a substitute for egg wash, particularly for savory food, especially fried chicken. That’s not surprising considering that mayonnaise contains egg. If you want to avoid eggs, try a vegan mayonnaise.
To use: Mayonnaise is thicker than a typical egg wash. You can thin it out by adding milk or water. Brush a light layer over the unbaked surface.
Soy Milk and Rice Milk
Soy milk and rice milk are not a typical choice for the glaze. Instead, they can work in place of a pre-batter dip, or combined with flour to make a sticky batter.
To use: Use as much as you need. You may need a thick layer as these kinds of milk are typically quite thin. However, avoid using a very thick layer as it can make your dish soggy.
For oven-baked food, try dipping the food in milk or yogurt then rolling in the breading or crumbs. The dairy will help bind the breading while retaining the moisture. 
To use: If the yogurt is thick, you can thin it out by adding a little water. Use this just before breading.
While baking bread delights, using a malt syrup as an egg wash alternative can give your dish a nutty and caramelized flavor. It is often used to glaze meat, such as pork while roasting. Its mellow sweetness pairs well in savory dishes as well.
To use: Malt syrup is quite thick and hence, the resultant glaze will also be thick. Use a pastry brush to glaze the meat. If using on a pastry, you may want to thin it with a little water before using it.
It goes best with fruit-based baking. You can also pair it with ham or salmon. The smoky-sweet flavor of molasses pairs very well with scones as well.
To use: It is best to use medium or dark molasses for glazing meat. For other baked food, you can also use light molasses. It’s best to avoid backstrap. It has a bitterness for glazing or as a substitute for egg wash as it has a bitter edge. Use only where the recipe recommends.
A combination of custard powder and water can give your pastries the perfect finish, making them look appealing. It gives a smooth and lustrous finish.
To use: Dilute the custard powder in equal amounts of water. That is, for each teaspoon use one teaspoon of water to make a paste. Apply this with a pastry brush.
This is another great option for vegans, particularly as binding before breading. Ground flaxseed can work as a great substitute for beaten eggs. When combined with water, it is known as a flax egg and used in place of egg in vegan baked products.
To use: Mix one tablespoon of ground flaxseed with three tablespoons of warm water. Let it rest for 10 minutes as it thickens.
If you want to try out more options, you can use aquafaba, agave syrup, and many butter substitutes like coconut butter.