This dish can be made with one or both types of scallops.
Heat oven to 450°. Lay bay scallops, in their shells, in a shallow baking pan and roast about 5 to 8 minutes, until scallops become opaque and shells have opened. Remove scallops from oven and transfer to a plate. Allow to cool for a few minutes. Cut each scallop away from its top and bottom shell and trim all material away from the fleshy white muscle. Place scallops in a small dish, then pour pan juices over the top. Chill, refrigerating the shells as well.
In a shallow bowl, toss sea scallops with oil to lightly coat, then season with salt and pepper. In a medium skillet with a heatproof handle over medium-high heat, cook scallops about 2 minutes on each side, or until browned; do not move them. Remove skillet from stovetop and place in oven about 5 minutes. Transfer to a plate and refrigerate at least 1 hour.
Place carrots in a small saucepan with a lid. Fill with just enough water to meet the top of the carrots. Add ginger. Cover and bring to a simmer until carrots are tender, about 10 minutes. Pour into a bowl and let cool. Once cooled, strain liquid and reserve. To make vinaigrette, place carrots in a blender with vinegar and about half of the reserved liquid; puree until smooth. Add more liquid if needed and season with salt to taste. Chill.
To serve, toss scallops with cilantro, scallion, and lime. Spoon about 1 tablespoon of the carrot vinaigrette onto a scallop shell. Place a sea scallop in the center and arrange bay scallops around the edge.