Why do doughnuts have holes?
Legend has it that the inventor of the ring shaped donut with a hole came up with the idea because he was dissatisfied with the raw centre of regular donuts.

http://en.wikipedia.org/wiki/Doughnut#History_in_the_U.S.

Having done quite a few doughnut experiments recently, it's true that having shapes where the distance from the centre to the surface is smaller increases your chances of properly cooked doughnuts. So toroids make sense, but then so would thin cylinders.

You can however, fry up donuts will no hole, that are cooked all the way through. Look at Berliners or jam-filled doughnuts.