Place the monkfish fillets on a work surface. If previously frozen, ensure they have thawed completely before proceeding.
Pat dry with paper towels to remove excess moisture. If skin or the grey membrane is present, use a butcher's knife to remove it. Slide the blade beneath and work it back and forth to trim off the membrane/skin (optional but recommended).
If the fillets are large, cut in half lengthwise down the middle to create strips 1-2 inches thick (2 1/2 - 5 cm). This step will help them to cook quickly and evenly.
Rub the fillets all over with 1 tablespoon olive oil. Then season generously with salt and pepper.
Place a nonstick skillet oven medium-high heat. When hot, add the butter and remaining 1 tablespoon olive oil to the pan. Swirl the pan to coat.
Add the monkfish fillets to the pan. Fry undisturbed for 3 minutes to let the fish start browning.
Using a fish spatula or kitchen tongs, carefully flip each fillet. Spoon pan juices on top to baste the fish. Continue cooking as little as 3 minutes more for one-inch fillets and up to 12 minutes for large fillets.
Check doneness by inserting an instant-read thermometer into the center of the fillet. It will be rare at 125°F (52°C), medium at 135°F (57°C) and well-done at the USDA safe temperature of 145°F (63°C). If you don't have a thermometer, slide a knife tip 1/4-inch into the thickest section and twist slightly to expose the flesh. If it's translucent and flakes easily, the fish is done.
Remove the fillets to a plate. Use a half lemon to sprinkle droplets of juice onto the fish along with optional parsley. Drizzle pan juices on top as well. Serve warm with lemon wedges.