In a dish large enough to hold the scallops in a single layer, whisk together the miso, mirin, sake, 1/4 cup minced scallions, and ginger. Add the scallops and turn them so they are well coated with the marinade. Cover with plastic wrap and refrigerate for at least 2 hours; if you can plan to marinate the scallops overnight, all the better.
Preheat the broiler. Oil a rimmed baking sheet, or spray with nonstick cooking spray (you can also line it with parchment for easier clean up). Remove the scallops from the marinade. Place the scallops on the baking sheet, with at least 1/2-inch between each piece. Broil for 3 to 4 minutes, until golden brown on top and just cooked through inside. You do not have to turn the scallops — get one side nice and brown. Make sure the middle is just cooked through; the interior should be opaque, a creamy white, with no translucence in the center.
Sprinkle the scallops with lime juice and the sliced scallions, if using, and serve hot, browned side up.