First, a word on handling and preparing stingrays for those who have never bothered with them before, it goes without saying to BE VERY CAREFUL with the barb. Before you do anything else, chop off the tail near the body with a long sharp heavy knife, and carefully dispose of it somewhere that you or anyone else (including a dog nosing through garbage) might possibly absent-mindedly run into it again.
Also, it is vital that you fillet stingray as soon as possible after you catch them (within 6 hours) as they start to produce ammonia after they die.
The best size stingray has a wingspan between 18 inches and 2 feet. Any bigger and they tend to be tough and stringy, any smaller and they aren't worth messing with. If you're at all proficient at filleting flounders, you're in luck, because the procedure for filleting stingray wings is exactly the same. With the ray lying flat on the cleaning table, poke with your finger to find the line where the tender wing joins the hard back. Using a sharp knife, slice downward along that line from front to back, just deep enough to reach the cartilage.
Next, turn the knife blade flat and work it along the top of the cartilage out toward the wingtip—just like separating the fillet from the bones of a typical fish. Do the same for the opposite wing.
Now turn the ray on its back and fillet the undersides of both wings in the same manner. The bottom fillets will be thinner and smaller—but good. The final step is to skin the four fillets by running the knife between flesh and skin.
Now with that taken care of, you can move onto the cooking. In addition to your stingray wings (assuming you started with one 18-24" wide ray) you will need:
1/4 cup all-purpose flour
Salt and fresh ground pepper
4 tablespoons of unsalted butter, divided
Juice of one lemon
1 tablespoon small (nonpareil) capers, drained
1/2 cup small croutons, plain
1 tablespoon flat leaf (Italian) parsley, chopped
Season the flour with the salt and pepper to taste. Dredge the ray wing fillets in the flour mixture, shaking off any excess flour.
In a large saute pan, melt 1 tablespoon of the butter over medium-high heat. When the butter has foamed and settled down, add the fillets and cook on high heat for two minutes. Add another tablespoon of the butter, flip the fillets, and cook another two minutes. Transfer to a serving platter.
Pour the butter out of the pan and wipe the pan out, then add the remaining 2 tablespoons of butter. Cook over high heat until the foam just settles down, then add the capers and croutons. Cook for 30 seconds. Add the lemon juice and parsley. Remove from heat, and spoon onto the fish.
This goes well with or on top of a plain pasta like linguine, and simple vegetables like a sauteed broccoli rabe or broccolini (sauteed in olive oil, garlic, salt, a squirt of lemon juice). I'd pair it with a citrus-forward medium-bodied wine like a New Zealand sauvignon blanc or Italian gavi.