Remove the monkfish from the refrigerator (or thaw completely if frozen). Pat dry with paper towels and rub with 1 tablespoon olive oil on all sides. Season with salt and pepper.

Place a heavy skillet over medium-high heat. When hot, add the remaining olive oil and butter to the pan and swirl the pan to coat.

Add the monkfish and fry undisturbed for 5 minutes. Using a fish spatula or kitchen tongs, flip the fillets and cook several minutes more, spooning pan juices on top from time to time. Note that 1-inch thick fillets will take 6-7 minutes in total, while 2-3 inch thick fillets could take 15 minutes or more.

To check doneness, insert a knife tip into the thickest section for a moment. If it comes out hot to the touch, then the fish is done. You can also insert an instant-read thermometer and look for a reading of 145°F.

Sprinkle lemon juice and optional fresh parsley onto the monkfish. Transfer to serving plates immediately and drizzle pan juices on top.