Sprinkle both sides of the fish fillets with salt and pepper.
Heat the oil and 1 tablespoon butter in a large nonstick skillet over medium high heat. When the fat is sizzling, add 2 fillets and cook until golden brown on the bottom, about 2 minutes. Flip and cook until golden brown on the other side and the fish is cooked through, about 2 minutes more. Remove the fillets to a serving platter, leaving any fat behind in the skillet. If the skillet looks dry, add another tablespoon of oil. Repeat the cooking process with the remaining 2 fillets and transfer to the serving platter.
Add the shallots to the drippings in the skillet and cook over medium heat until softened slightly, about 2 minutes. Add the capers and continue to cook until the shallots have completely softened, about 2 minutes more. Add the wine and lemon slices, bring to a simmer and cook, swirling the pan occasionally, until the liquid has reduced by half. Add the lemon juice and remaining 4 tablespoons butter, swirling the pan to emulsify the sauce as the butter melts. Stir in the parsley and season with salt and pepper. Spoon the sauce and lemons over the fish and serve immediately.
You can substitute any type of mild white fish in this recipe, though thin filets are recommended. Flounder, tilapia and whiting are all good alternatives
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