Combine the cilantro and oil in a pan and warm over low heat for about 10 minutes, until fragrant. The oil should not simmer.
In another pan, heat some oil until smoking. Season the fish and carefully place it in the hot pan. Let the fish sear until deeply browned, about 1 minute. Flip the fish over and repeat on the other side until done. (If your fillets are thick, you should lower the heat on once the first side is seared.)
Remove the fish from the pan and keep warm. Pour out oil so that only 1 tablespoon remains in the pan.
Add the butter lettuce to the pan. Cook until bright green and wilted.
To plate, lay the butter lettuce down in the center of each plate. Place the fish on top with the best sear facing up. With a spoon, drizzle the cilantro oil around the fish and on top of the lettuce.