Pat fillets dry. Dust with flour, and shake off any excess. Sprinkle them on both sides with salt and pepper.
Prepare a heavy pan with the olive oil and 1 TB butter. Heat on medium-high until butter is melted but not browned, then turn the heat down to medium.
Place the fillets in the pan and cook 2-4 minutes. Don't touch the fillets until they're ready to flip.Note: Thin fillets like flounder will need to cook approx. 2 minutes per side, Dover sole 2-3 min per side, cod + halibut 4 min per side.
Use a thin metal spatula to test the fish. If you can slide the spatula under the fish without the fish sticking to the pan, it's ready to flip. If the fillets won't release- give them another 30 seconds. They'll release when they've finished cooking.
Cook fish another 2-4 minutes until done.(See note above for cook time.)
Divide the 1 TB butter over both fillets. Transfer the fish to plates, leaving the excess butter in the pan. Cover fish to keep warm, or set fish in a 200ºF oven.
Reduce heat to medium. Add the wine to the pan, and allow it to reduce for about a minute. Add the remaining butter and lemon juice and stir with a wooden spoon. Add half the herbs and stir again.
Optionally, cut remaining lemon in half. Turn heat to high, and set lemon cut-side down on the hot pan. Char 2-3 minutes until the cut side is charred.
Serve fish hot with pan sauce, remaining herbs, and optional capers and lemons.