I picked up a collection of new nipples and matching nuts at Home Depot today and brought them home to use in adjusting the fixture attached to the ceiling fan in the office so that it would hold two of the Cree 100W LED bulbs with the cover on. Later in the afternoon, I hit the old nipple with some penetrating oil, waited a bit, and tried to turn the nuts with a pair of pliers. No luck.
So daisy_knotwise got me the Vicegrip and I got the socket set from the basement. Between them, I was finally able to loosen the nuts and retrieve the mounting plate from the old nipple. I threaded it onto the new, slightly longer nipple, made some fine adjustments, and got the cover back on.