Place the butter in a medium pan over medium-high heat. Let the butter melt and slowly turn brown. Make sure not to let the butter burn, and adjust the heat as needed.
Season the fish filets with salt and pepper. In a medium mixing bowl, combine the flour and seasoning salt.
Dredge the filets in the flour mixture, coating them evenly on each side. Set the fish aside until ready to use. When the butter is golden brown, add the garlic and anchovies. Sauté for a few minutes until the anchovy has dissolved and the garlic is fragrant.
Add the fish to the pan, skin side up. Fry on each side for about 3–4 minutes, or until a golden crust has formed.
Add the white wine and bring it to a boil. Let the mixture reduce for a few minutes, then cover it with a fitted lid and reduce the heat to medium-low. Allow the fish to cook in the sauce for 15 minutes, or until it’s white throughout.
Serve immediately with extra brown butter sauce and enjoy!