Preheat 2 medium non-stick skillets over medium-low heat for 5 minutes.
Meanwhile, lay the fillets on a piece of wax paper, parchment, or a baking sheet and blot dry with paper towels. Season the fillets with salt and pepper. (If the fillets tapers down to a very thin tail, fold the tail back to make a more uniformly thick fillet.) Put the flour on a plate. Dredge the rounded-sides of the fillets in the flour and shake off any excess.
Raise the heat to medium-high and add 1 tablespoon of butter to each skillet. When the butter stops foaming, add 2 fillets to each pan, rounded-side down, and cook, lightly shaking the skillets occasionally, until golden brown, about 2 minutes. Turn off the heat, flip the fillets with a spatula, and let the fillets finish cooking in the residual heat in the pans, about 2 minutes more. Transfer the fillets to 4 warm plates and cover loosely with foil.
Add the lemon juice and water to one of the skillets. While moving the skillet in a circular motion, add the remaining 3 tablespoons butter, little by little, until melted and a smooth, creamy sauce is formed. Stir in the parsley and thyme and season with salt and pepper to taste. Spoon some of the sauce over each fillet and serve immediately.
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