Hi Redunning1! I'm still working on a final egg wash recipe but so far I'm using a mixture of 50% soy milk and 50% amber agave syrup or corn syrup. Note that this doesn't result in as much glossiness as a true egg wash.

The soy milk contains protein which encourages browning and the amber agave contains sugar which caramelizes. I used it on my Chocolate Croissants just to give you an idea how it works.
http://veganbaking.net/pastries/717-chocolate-croissants

If you want additional browning I recommend blending the mixture with about 1/2 teaspoon baking soda. Alkaline solutions promote browning, such as with baking soda and acidic solutions discourage browning. An egg has minor browning properties due to it being more on the alkaline side.

Good luck!