Chop the basil and garlic together on a cutting board as fine as you can manage, almost a puree (a mini food processor or mortar and pestle will help you here).
Make a deep horizontal slit in the side of each scallop but don’t cut all the way through. Fill each scallop with about 1/2 teaspoon of the basil paste. Pour the remaining oil onto a plate or pan and turn the scallops in it. Let them sit while you turn on the broiler and position the rack about 4 inches below the heat or prepare a charcoal or gas grill for hot direct cooking; make sure the grates are clean.
Remove the scallops from the oil and put them under the broiler or on the grill (don’t pour the remaining oil over them, as it will catch fire) and cover. Broil or grill, turning once, until golden all over and the stuffing is warm, 2 to 3 minutes per side, no more. Serve hot or at room temperature with lemon wedges.