Place the flour on a plate. Season the sole with the salt and then coat it in the flour, shaking to remove any excess; set aside.
Melt 1 tablespoon of the butter in a large skillet over medium heat. Slice the lemon into 12 thin circles and add them to the skillet. Cook until the lemon is lightly browned, about 2 minutes. Push the lemon to the side of the skillet and add the sole. (You may need to cook it in batches.) Cook until the sole is the same color throughout and flakes easily, about 2 minutes per side. Add the remaining butter and the capers to the skillet. Remove from heat and tilt the skillet to swirl the butter until it melts. Transfer the sole and lemon to individual plates and spoon the capers and butter over the top.
Tip: If cod isn't available, you can substitute another delicate, mild-tasting white fish, like cod, red snapper, or tilapia.
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