This can be made in any wood which can be turned on a lathe.
You will need a block of wood 4 1/4" x 4 1/4" x 9".
Print out the shape on the next page. Next do the base cutting the inside lip before you finish the outside. You can then fit the lid to make sure you get a perfect matching slope between base and lid.
Finish the base with a small flat bottom as shown so that the egg will stand vertically.
If you have to build it up from several pieces make sure you get excellent joins and of course surface to surface, no joints, dowels or biscuits.
Copy the shape on to the block.