How can I cover an egg-washed dough?
I'm assuming the shaped dough is either on a flat surface or in a bread pan of some sort?

In a pinch, the simplest thing is often an inverted large bowl or similar container.  That will prevent air circulation.  In general, dough is mostly covered during the final rise to prevent a "skin" forming if the outer layer of the dough dries out.  (A skin can be a hindrance to a complete rise and/or impede oven spring, not to mention potentially changing the texture of the final crust of the bread.)  With the combination of an egg wash (which already moistens the dough) and a bowl or other container inverted to prevent air circulation, that's probably enough to prevent the exterior from drying out too much.

If you're still concerned about keeping the exterior moist, the next level is to use a larger inverted bowl or container that's big enough to allow a small cup of hot water to sit beside the dough underneath it.  Unless it's a very long rise, the initial heat from the cup will humidify the air and prevent the surface of the dough from drying out.  (For longer rises or in a cold kitchen, you can refresh the cup with more hot water periodically.)  Another option I've heard that some people use is to put the dough and the hot cup of water in a microwave and shut the door.  The microwave isn't turned on; it's just used as a relatively small somewhat sealed area.  You don't usually need something perfectly sealed or that fits the dough precisely -- as long as you prevent a lot of air flow around the dough, it's often enough to prevent a skin from forming.

The cup of hot water step probably isn't necessary when using an egg wash, but it's a way to keep dough from drying out in general, simulating a "proofing box" as used by professional bakers.  (Professional proofing boxes usually have more precise temperature and humidity control, however.)  After doing these sorts of tricks for a couple years, I eventually cut the top flaps off a larger cardboard box, taped up the bottom to seal it, and used that as an inverted makeshift "proofing box" which was big enough for a few loaves.  You could also buy a larger plastic bin or something for this purpose.  If you do things like this often, it may become worthwhile to invest in a small electric proofing box that allows you to control temperature more precisely.  (I was given one as a gift one year, so that's what I now use for this purpose.)