The joke is basically a riff on how opossum is considered at least by some people to be inedible, or at best the sort of thing that only the most backwoodsy of backwoods people would consider eating. The truth (and here there were far more Google results) seems to be that, because they're scavengers, the flavor is highly variable depending on what they've been feeding on.
Anyway, so the joke. The recipe for planked possum is that you start by taking a plank, and you attach the possum to the plank, and roasting it. This should generally be told with lots of culinary details and flourishes: A fresh-cut cedar plank, layers of cut onions between the possum and the plank, that sort of thing. Then, once this has roasted for an unreasonable length of time, you take it out of the oven, and for the punchline you discard the possum and eat the plank.
The "Interview with Jack S. Baird and Richard S. Potter, 1982" from UTSA's Oral History Collection has a really nice example of the genre, which seems to have taken inspiration from the interview being held at the East Texas Yamboree -- a folk festival honoring yams. ("M" is the interviewer, "P" is the interviewee, and they're talking generally about things that he's talked about at his "Possumology" show at the festival and here specifically about recipes.)
M: Tell [me] about the Possum Treat.
P: Well, you get about a 5, 6 pound possum, clean it good and get most of the fat tissue off of the carcass. Then you parboil it and you get about 6 East Texas yams and you cut them lengthwise. Then get half, three quarter[s of a] cup of good syrup, [from] ribbon cane, East Texas ribbon cane, and you bring it to a boil. Boil it about 4, 5 minutes, turn it off ... just keep it on warm. Then you get a white oak board ... that's what most of 'em use ... and it's usually an inch thick and 12 by 18 inches in length. You put the possum in the center and you surround it by the yams.
M: Have the yams been baked?
P: The yams have been parboiled. Possum has been parboiled as well. To finish it and get the sweet syrupy taste, you pour some of the syrup on it, put it in a 250 degree oven and cook it for about 2 hours. About every 30 minutes, baste it with the ribbon cane syrup ... on the possum and the yams. And at the end of 2 or 2 1/2 hours, you remove it from the oven and go to the nearest garbage can and scrape off the possum and the yams into the can and then you eat the board. It's a delightful flavor.
So there you go. Now maybe there are three findable references online.
Crossposted from Dreamwidth (original here), with comments. Comment here or there; comments here will eventually be duplicated to there.