Preheat the oven to 300° F.
Dry the sole fillets and season both sides with 3/4 teaspoon of the salt and the white pepper. Dredge the fillets in the flour and shake off the excess.
Heat 2 tablespoons of the oil in a large nonstick sauté pan over a medium-high flame. When the oil smokes, add 2 fillets and cook for 3 minutes on each side.
Transfer to a baking sheet, place in the warm oven and continue with the next 2 fillets. Add oil as necessary, heating to the smoking point before adding the fillets. Repeat with the remaining 2 fillets.
Transfer all the cooked fillets to the oven and wipe the sauté pan clean with a paper towel. Add the butter to the pan over a medium flame. It will melt, foam, and subside, and then begin to bubble again and turn a nutty brown.
Add the lemon juice, parsley, the remaining ½ teaspoon of salt, and several grindings of white pepper to taste. Serve the fillets on warm plates with the sauce spooned over.
The Johnson Family Cookbook