Preheat the oven to 450℉, then place a cast-iron skillet in your oven for 20 minutes.
Pat the filets with a paper towel to remove moisture and season liberally with salt and pepper.
At the same time, bring a pot of salted water to a boil.
Turn off the heat under the boiling water, and add the lobster meat to the pot and cook for about 1 minute. Using a slotted spoon, remove and set aside.
Transfer the hot skillet carefully from the oven to the stovetop using a thick oven mitt.
Coat the bottom of the skillet with a neutral oil, like safflower, and set the temperature to high heat.
Use tongs to place each filet in the pan for 30 seconds on each side.
Add butter to the pan with the steaks and place it back in the oven.
Cook steak for 2 minutes on each side for medium rare. Add 2 minutes for medium. A medium-well steak should take about 8 minutes in total. Remove the steaks and put them on a plate to rest.
Quickly transfer the lobster meat to the skillet, which should still be hot. The lobster is finished when it becomes tender and is opaque inside. Squeeze the lemon juice over the lobster meat.
Place each rested filet on a separate plate and garnish each with a quarter of the lobster meat, and season lightly with black pepper. Serve right away.