Preheat oven to 200 degrees (to keep cooked scallops warm for a few minutes while preparing lemon butter sauce). To prepare the scallops, examine each one for a side muscle, a square-shaped bit of tissue that runs in the opposite direction as the rest of the muscle. Pull it off by pinching it and tearing it away from the scallop.
Line a rimmed baking sheet or plate with a clean kitchen towel. Arrange scallops in a single layer on towel, then cover with a second clean kitchen towel. Press down gently to blot liquid.
Let scallops rest at room temperature until towels have absorbed most of their moisture, about 10 minutes. Season both sides of scallops with salt and pepper to taste.
In a large skillet over high heat, heat 1 tablespoon vegetable oil until just smoking. Add half of the scallops in a single layer, flat-side down. Cook, without moving, until well-browned, about 1 1/2 to 2 minutes.
Add 1 tablespoon butter to the skillet. Using tongs, flip scallops. Continue to cook, using a large spoon to baste each scallop with butter until the sides of scallops are firm and centers are opaque, about 30 to 90 seconds longer (tilt skillet to one side so the butter pools and is easier to scoop). Remove smaller scallops as they finish cooking.
Transfer scallops to a large plate and tent loosely with foil. Wipe out skillet with paper towels and repeat with remaining vegetable oil, scallops, and butter. Transfer to oven to keep warm while preparing lemon butter sauce.