To get the bread crumbs to stick to the chicken, I coat the chicken with eggs. I then put the bowl in the refrigerator. How long should I wait before taking the chicken out of the eggs and rubbing the bread crumbs on them?
Spices or egg do not get absorbed into chicken, or any other protein in any significant manner given any safe amount of time
Some amount of sodium from a brine will get absorbed, but this is generally for different reasons, and is not a requirement for crumbed chicken
The easiest and safest method is to dredge in flour, dip in egg, dredge in crumbs, and cook (deep fry, pan fry, oven etc) all in one sequence of steps
The flour sticks to chicken well, and become a glue to hold the egg when it gets wet and hot during cooking. Egg alone will not stick well to chicken, see this post. The egg will stick to the flour well, and bread crumbs or Panko will stick to egg well see this post
For food safety reasons prepared foods should not be sitting around for too many hours. The normal process is to prepare and then cook. Also you should not save flour of crumbs for another day see this post
In my experience; it is best to allow the chicken to warm up a bit after being removed from the fridge, e.g. rest on bench 30 minutes before you plan to cook. Make sure the chicken is not wet, damp is OK. Line up your flour, egg, and crumb mixtures a short distance away from your cooking device of choice (for speed, simplicity, and safety). And in one continuous process; flour dredge, egg dip, crumb dredge, and then cook
Any waiting will generally cause coating failures. You do not need to be fast, but you don't want to have it sitting around. You fingers and kitchen will be messy, so no point stopping, get it all done in one go
As I always see my mother, she just coat it with egg then apply breadcrumbs and directly place it in the hot frying oil. Waiting is just for spicing or seasoning to allow the meat catching spices flavors.