Remove the side muscle or tendon from the scallops before cooking them. It’s edible, but a tougher part of the scallop, and most people are happier with it removed. Place the scallops on a clean dishtowel or paper towels. Cover them with another clean dishtowel or paper towels, press the toweling gently onto the surface of the scallops, and let sit for 5 minutes.
Heat the oil in a large pan, preferably nonstick, over medium high heat. While the pan is heating, season the scallops on both sides with salt and pepper. Add the scallops without crowding them in the pan (cook in batches if needed). Cook, without moving, for 2 to 3 minutes until the bottom is nicely browned. Flip the scallops with a spatula and allow the other side browns, for about 1 to 2 minutes. While the second side is browning, add the butter to the pan and as it melts, use a spoon to baste the tops of the scallops.
Transfer the scallops to a plate (f you are making a sauce for the scallops, you can tent them with aluminum foil to keep them warm while you make the sauce). Drizzle the pan juices over the scallops and serve with the lemon wedges.
Notes
Dry the scallops well. Even if you are using dry-packed scallops (which you should be), you want to get them as dry as you can. Blot the scallops dry with a clean dishtowel or paper towels before cooking them. Removing as much moisture as possible helps with browning.
Salt just before cooking. Use kosher or flaky sea salt, not iodized salt. Don’t salt the scallops until just before cooking them. Slating them too early will start to pull out moisture from the middle of the scallops and impede browning.