I threw the square piece in the chuck, hollowed it out to a bit undersized, polished it to the correct size, then put the chuck on the inside to turn the outside.
I polished the outside on a jam chuck made from a chunk of broom handle.
Super fast and easy, though this is the smallest size I could grab from the inside with my chuck. I would need a smaller set of jaws to hold anything smaller.
To make the smaller rings I did for my niece, I put one end of the piece in the chuck, turned both the inside and the outside while attached, then parted the ring off the end.
I hand polished the insides and used the jam chuck to polish the outsides.