Lemontree said I should machine something fun. So I cut off a piece of brass, drilled a hole across it in the drill press, drilled again through the middle on the lathe, then cut a 45° angle across each end. Then I cut it in half and got two lamps:
The cross-drilled hole is for the wick and oil to get in to the central hole. Tiffany supplied the wicks from her crochet kit.
You're supposed to use olive oil because it doesn't smoke like vegetable oil will. We seemed to be out of olive oil, so a quick Google search revealed that sesame oil works as well. I'm pleased to report The Internet was right, no smoke or even soot on the back of a spoon placed over the flame. The lamps do produce a faint odor of sesame, though, which is pleasant enough... as long as the neighbors don't come to our house looking for the new Mongolian Barbecue.
We just put them on a saucer, but they get kind of hot so we should probably find a little candle jar with a thick bottom to put them in.