Easy, delicious and perfect for both, a satisfying weeknight dinner as well as for entertaining. I prefer grey sole fillets for this because they are delicate, yet firm, and seem to hold up well. If you need to, just substitute lemon sole, although you might need to halve the fillets before dredging to make them easier to work with. The batter for the fish is very simple to prepare-It’s like making a custard for french toast, only a bit more “eggy” and the addition of parmesan cheese makes it savory (Yes, you can have cheese with fish…)
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Rinse the fish and dry well. Place on a plate lined with wax paper and season both sides lightly with salt and pepper. Place, in assembly-line fashion, the seasoned flour, the seasoned eggs with milk parmesan cheese (using a pie dish is a good shape for the egg mixture). Have this, along with the fish fillets, close to the stove. Have a platter warming, ready to receive the fish.
Heat a large, nonstick skillet over medium-high heat with a very layer of olive oil. When the oil is hot, add 2 tablespoons of the butter and, while butter is melting (working with one fillet at a time) dredge the fish in the seasoned flour and smack off the excess. Dip into the egg batter, coating both sides (allowing excess to drip back in pan) and then lay in the hot fat in the pan. Try to work quickly to help fill the pan before the fat overheats (reduce heat if needed). Cook the fish until golden on both sides and fillets are cooked through (this happens pretty quickly with grey sole fillets). Using a large turning spatula, remove the fish fillets, as they are cooked, to the warmed platter. (If your skillet wasn't large enough to cook all fillets at one time, you might need to add some more fat to the pan, as needed.) Cover platter to keep fish warm.
Pour out fat from pan. Put the pan back over medium-high heat and add 1/2 stick butter. When butter is hot and bubbling, add the minced shallot and saute until softened and fragrant, then add the wine and lemon juice and bring to a brisk bubble. Reduce the liquid by about 1/2 and then add the capers, along with some chopped parsley and freshly ground black pepper. Spoon the sauce over the cooked fish and serve hot.