I purchased one of the Mega Eggs at Hobby Lobby. They are only there for a limited time so I suggest you buy one as soon as possible. Otherwise you'll have to wait until next year to do this project. The large mega egg measures about 16 inches in height. It's a bit smaller than the egg chair that American Girl offers. But it will work.
The first thing to do is make a template for the opening. I do not recommend just going commando and cutting an opening because the plastic is difficult to cut and you want to make the right cuts. So grab some wrapping paper, paper, or tissue paper and sketch out basically an egg shape roughly to the size and shape you think would work. Then place the template on the egg and tape it down in areas. Once you do that you can then examine the template and see if you like it, its the right shape and is big enough. You do not want to make it too small, but you also don't want to make it too big. If you make it too small you can always cut off more plastic. But remember the plastic is hard to cut so you want to try to do that only once.
Once you have the opening cut out you need to reattach the two halves. I used a white glue, then taped the inside of the egg with duct tape and shipping tape on the outside.Then let it sit overnight to dry. Since the inside is going to be covered I suggest just leaving the duct tape on. The next morning I took the shipping tape off and discovered parts of the seam did not bond. So I ended up using mod melts. Mod melts are a new craft from Plaid that involves a high temp glue gun and rods that are plastic. I ended up using that as a glue as the plastic rods melt and bond with other plastic fairly well. You could also try using a super glue or Gorilla glue.
Once you got the egg together I suggest working on the base. For the base I used a 7 1/2 inch diameter plastic plate I bought at the dollar store, a plastic cap from a medicine bottle and a bolt, nut and washer, and white spray paint.
Once the base is dry, you will drill a hole in the base. (See pic) as well as the bottom of your egg. Then run your bolt through the egg, the cup and the plate and secure with a washer then a nut. This will hold the base to the egg as well as give it some stability.