This is Tom's recipe. Tom is a long time family friend, an excellent cook, and has been and lived everywhere. Literally. All over the world. It is not really what I would call barbeque--it is not cooked outdoors, or on a grill, nor does it have a typical barbeque sauce. But Tom calls it B.B.Q. Shrimp and I'm sticking with that.
These are peel and eat shrimp, baked in the oven with a savory sauce. The sauce has intense flavor, quite a kick, and is delicious as a dip for french bread. The recipe calls for baking the shrimp for 30 minutes, then turn off heat and let them set in the oven for another 30 minutes. This seems like an inordinate amount of time to cook shrimp but I have made this recipe many, many times, and it works. The shrimp become completely infused with the flavors.
Because we don't have a lot of shrimp beds here in the mountains, we have to use frozen shrimp. We like the "E-Z-Peel" shrimp, which have already been deveined, that we find at Harris Teeter. But if you are lucky enough to get fresh shrimp, all the better. The first time I ate this dish it was made from fresh Gulf Coast shrimp. Sublime.
Here's Tom's recipe, in his own writing.