How to Make It
Rinse the scallops well, and scrub to remove any grit. Using a knife, carefully open each shell, and trim away the connective tissue that holds the muscle to the shell. Be sure to remove all matter in the shell except the actual muscle itself, which will be bright white and firmly attached to the shell. (Parts that should be discarded are darker, cloudy, gelatinous, and not attached to the shell.) Gently loosen the scallop muscle from the shell, leaving it in the half shell for grilling. Place scallops (resting in the bottom shell) over a bowl of ice while you clean the others.
Stir together the butter, scallions, garlic, lime zest, lime juice, salt, paprika, cayenne pepper, and 1/3 cup cilantro in a bowl. Shape mixture into a log, and wrap in plastic. Chill until firm, about 30 minutes. (This can be done up to two days in advance.)
Preheat grill to medium (350°F to 450°F). Place cleaned scallops on a tray or in a rimmed baking sheet. Top each scallop with a generous teaspoon of the cilantro-lime butter. Place scallops on the grill, and grill just until the butter starts to sizzle, 2 to 4 minutes. Using tongs, remove from heat, and let stand 2 minutes. (The scallops will finish cooking as they stand.) Serve immediately.