Announcing Poly Crepe de Chine!
Thanks to "inscribed_here", this is now available as a microwave heating bag*. You can trim the white edge from the yolk, or simply cover it with the stitching. Sew the yolk to the egg, using a zigzag stitch around the edge, without stuffing it (your egg won't appear fresh, but considering you will be using it over and over how fresh would it be anyway?) If desired, you can trim the egg to a more uniform shape. Choose any fabric you like pin the fabrics, right sides together. Trim the backing fabric even with the egg, sew around the edge, leaving an opening to add your filling.
Here is a site where you can find suggestions for fillers and instructions for using your egg: Tipnut; and thanks to "Tipnut" I now know that these can also be called: wheat bags, bed buddies, stress busters, magic bags, corn cozies, and rice bags.
Thanks to "connielou" this is now available as an infant costume. Instructions in comments below.