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Step 1
Melt butter in a light colored saucepan over low heat. Continue to cook at a low temperature until butter starts to foam and then begins to turn brown. Remove from heat as soon as it browns and transfer into a bowl. You may run it through a sieve when you transfer the butter to the bowl, if you desire. Place a sprig of rosemary in the bowl and the lemon juice and let sit.
Step 2
Pat fish down with a paper towel. Salt and pepper fillets on both sides. In a flat bottomed dish big enough to fit a fillet, sprinkle flour evenly. Place fillets in the dish one at a time, flipping on both sides to ensure they are evenly coated with flour. [Optional. Fillets can also be prepared without the flour]
Step 3
Heat a couple tablespoons of olive oil in a skillet. Place fillets in the skillet. Cook on both sides for about 4 minutes or so. Cooking time will vary depending on the thickness of your fillets.
Step 4
Spoon browned butter sauce over fillets and serve.