Heat a heavy pan over medium-high heat.
Meanwhile, season the fish on the meat side with salt and pepper. Lightly dredge in flour and shake off excess.
When the pan is good and hot, add the oil followed immediately by 1 tablespoon of the butter. As soon as the foaming subsides, place the fish in pan with the skin side down. Jiggle the pan for the first 10 seconds of cooking to keep the fish from sticking, then continue to cook until the skin turns golden brown.
Carefully turn the fish away from you and again jiggle the pan for the first few seconds. Continue to cook until the flesh side turns golden brown. Remove to a warm plate.
Pour out the cooking fat, then add the remaining butter and the capers. Cook until the capers begin to crisp, then remove the pan from the heat, add lemon juice to pan and swirl. Pour the sauce over the fish and serve.