From America's Test Kitchen
Fish meunière typically features pale, soggy fillets in pools of greasy sauce—that is, if the fish doesn’t stick to the pan or fall apart as it is plated. We wanted perfectly cooked fillets that were delicately crisp and golden brown on the outside and moist and flavorful on the inside, napped in a buttery yet light sauce.
Whole Dover sole is the most authentic choice, but it’s also hard to come by and prohibitively expensive; either sole or flounder fillets are good stand-ins. To prevent the likelihood of overcooking the fish, the fillets need to be no less than 3/8 inch thick. The fillets must be patted dry before being seasoned with salt and pepper and dredged in flour (no need for eggs and bread crumbs). Using a nonstick skillet for pan-frying meant there was less chance for our fillets to fall apart; lubricating the pan with a mixture of oil and butter added extra insurance. Removing the pan from the heat just before the fish was cooked prevented the fish from being dry (the fish will continue to cook off the heat). Butter browned in a traditional skillet (so the changing color is easy to monitor) and brightened with lemon juice made the ideal accompaniment to our crispy, golden fillets.
Try to purchase fillets that are of similar size, and avoid those that weigh less than 5 ounces because they will cook too quickly. A nonstick skillet ensures that the fillets will release from the pan, but for the sauce a traditional skillet is preferable because its light-colored surface will allow you to monitor the color of the butter as it browns.