1. Rinse the fish and dry the surface and the cavity of each thoroughly with paper towels. In a 12-inch or larger nonstick frying pan, heat 3/8 of an inch of cooking oil over moderately high heat until very hot. A deep-fat thermometer should register 375°. Carefully add the fish, letting the tails stick out of the pan if necessary. Let the fish cook, without moving them, until crisp and browned, about 9 minutes. Using a large spatula, carefully turn each fish. Continue cooking until crisp and browned and just done, about 7 minutes longer. Drain on paper towels.
2. Meanwhile, in a small glass or stainless-steel bowl, combine the broth, soy sauce, lime juice, orange juice, sesame oil, grated ginger, and scallions.
3. Serve the fish on a platter. Run a knife between the flesh and the bones and lift off the fillet with the skin attached. Turn the fish over and repeat. Pass the sauce.
Fish Alternatives: In place of the whole red snappers, you can use whole sea bass, porgies, or rockfish of the same weight.
Wine Recommendation: A crisp, acidic white wine will be best with the citrus flavors and Asian ingredients of this dish. Try a sauvignon blanc from the Loire Valley in France (Sancerre) or a pinot blanc from the Alsace region.