Pat dry and season: Pat cod fillets very dry and season with salt and pepper.
Sear the cod: Heat olive oil and 1 tablespoon of the butter in a large non-stick skillet over medium-high heat until hot and sizzling. Add cod fillets in a single layer and cook, undisturbed, until golden brown on one side, about 2 minutes. Reduce heat to medium and, with a thin spatula, flip each cod fillet and cook 2 to 3 minutes more until nearly done (they'll continue to cook on the plate and be returned to the pan for 1-2 minutes, so take care not to overcook). Transfer cod fillets to clean plate.
Make the sauce: Add white wine, lemon juice, and capers to skillet and bring to boil; continue boiling until reduced slightly, 1-2 minutes. Add remaining 3 tablespoons butter and stir to melt.
Finish and serve: Return cod fillets to skillet and spoon sauce over until coated and heated through, 1–2 minutes more (or until cod is cooked to your liking). Remove from heat and transfer cod fillets and sauce to platter. Garnish with parsley and lemon slices if desired and serve.
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Notes
Pat fillets really dry: Moisture keeps the fish from browning properly, so don’t rush this step. It makes all the difference.
Give thicker fillets a little extra cooking time: If your cod is more than ¾ inch thick, add 30–60 seconds per side before finishing in the sauce.
Fresh lemon juice is worth it: Bottled works if you’re in a pinch, but fresh lemon gives the sauce its clean, bright flavor.
If the sauce tightens up: Stir in a teaspoon or two of warm broth or water to loosen it right before serving.
Reheat gently: Warm leftovers in a skillet over low heat until just heated through. Microwaving tends to make the fish tough.
Other fish that work well: Halibut, haddock, and sea bass all hold up nicely with the same method.