A couple of thoughts:

Non-diastatic malt powder gives baked goods better flavor and a shinier, browner crust. It's used in everything from bagels to croissants to breakfast cereals. It can be found online at King Arthur.

In addition I use a butter flavor cooking spray (I like the Crisco brand), spray on before baking and again right out of the oven – attached an example of a bread I baked using the cooking spray...crisco brand gives shine and a nice extra flavor.

As an egg-allergic person I would also love to see your recipe!

Also, can you post your vegan challah? For vegan friends, I adapted an eggless challah from The Blessing of Bread by Maggie Glazer, but always am looking for more options.

Maple syrup!

Have you tried sugar and water (brushing it on toward the end of baking)?

Have you tried soy products? Milk or cream for example?