I wish I had thought of this myself. I like the idea that the machine will draw neatly on an unusual 3D surface and that the result might make a unique gift. This machine does a great job despite the fact that every egg has a slightly different size and shape:
You can build the above model if you have the retail version of the EV3 - you can find the building plans HERE. It can probably also be made with the education version but you will have to modify the building plans.
Here's another machine, this one based on the NXT but with no building instructions:
But it's not a very interesting pattern - maybe a good challenge would be to have the machine write text on the egg or draw a face? And maybe it could draw on ping pong balls instead of eggs - ping pong balls have a more consistent size and shape and that might allow for more precise drawing?