2 large sole fillets, each 5½ to 6 ounces, rinsed and blotted dry
Salt and freshly ground black pepper
Unbleached all-purpose flour, for dredging
2 tablespoons unsalted butter, at room temperature
1 tablespoon freshly squeezed lemon juice
1 teaspoon small capers in brine
1 /2 teaspoons finely chopped flat-leaf parsley
Preheat your oven to 200°F. Put a plate in to warm.
Season the fillets with salt and pepper; dredge them in flour, patting to remove the excess.
Heat a 10-inch cast iron skillet over high heat until hot but not smoking, 3½ to 4 minutes. Add 1 tablespoon of the butter, swirl to cover the bottom of the pan, then immediately add the fillets and sauté for 2 to 2½ minutes, depending on the thickness of the fish, until golden brown; turn and cook the other side until lightly browned and just cooked through, about 2 minutes. Transfer the fish to the plate to keep warm.
Add the remaining butter to the skillet and cook until light brown; stir in the lemon juice and capers, bring just to a boil, and turn off the heat. Stir in the parsley, and season to taste with salt and pepper. Spoon the lemon-caper butter over the fish, and serve immediately.