1. If necessary, skin the fish: Start by cutting off the head. Then make a little V-cut at the tail in order to loosen the skin enough to get a firm grip on it. peel the skin back far enough to get your thumb under the loose skin. To prevent slipping, hold a towel in your hand as you grip the skin. In one strong, swift motion, pull the skin away from the tail—it should come off in one piece. Flip the fish over and remove the skin from the other side. Then trim the small bones away from the sides of the fish, and remove the roe sac (if any) and any viscera from the cavity. If necessary, cut off the tail of the fish so that it will fit into your pan. Rinse and dry the fish. Keep refrigerated until you are ready to cook.
2. Adjust a rack to the lower third of the oven and preheat the oven to 400°F.
3. Place a 12-inch well-seasoned skillet over high heat and heat for 5 minutes. Meanwhile, spread the flour in a large shallow bowl or a baking dish. Sprinkle the fish generously with salt and pepper and dredge it in the flour, turning it well to coat; shake gently to remove the excess.
4. Add the oil to the hot skillet. Lower the fish into the skillet and reduce the heat to medium-high (at this point, you should start cooking the butter; see step 5). Brown the fish on one side without turning, about 4 minutes. Turn the fish and immediately place the skillet in the oven. Roast until the fish is pure white and firm to the touch, about 8 minutes. Transfer the fish to a platter and keep warm.
5. Meanwhile, as soon as the fish goes into the pan, place the butter in an 8-inch skillet and melt it over medium-low heat. Reduce the heat to low and cook the butter gently until it browns—keep an eye on it, but let it cook undisturbed until it is nutty brown. This will take about 8 minutes (about the same time it takes to cook the fish). When the butter is nutty brown, remove the skillet from the heat and immediately add the lemon juice, capers, parsley and a pinch of salt. pour the still-foaming butter into a sauceboat or serving bowl and spoon just a bit over the fish.
6. Present the fish at the table and allow a minute for your guest to admire it. To fillet the fish, hold an ordinary tablespoon upside down the center of the fish, near the head, and loosen the top fillet, pushing out from the center of the fish. The bones on a flounder are very strong and won’t pull away with the meat. You should have no problem removing the two fillets on the top side. Transfer them to a dinner plate. Flip the fish over and repeat with the two remaining fillets. Spoon the brown butter over the fish and enjoy this special treat, noticing how much more flavorful fish is when cooked on the bone.