Coarsely shred the potatoes and squeeze out any excess liquid. Transfer to a large bowl and stir in the egg, matzo meal, onion and a large pinch of salt. Shape the mixture into 20 scallop-size cakes, about 1/2 inch thick; press the cakes to compress them.
In a large skillet, heat 1/4 inch of vegetable oil until shimmering. Working in batches, fry the pancakes over moderately high heat until browned and crisp, about 3 minutes per side. Lower the heat if the pancakes brown too quickly and add more oil if necessary. Transfer the pancakes to a baking sheet and sprinkle with salt.
Preheat the oven to 325°. In a small saucepan, combine the Champagne with the peppercorns, thyme sprigs, bay leaf, shallot and lemon juice and boil over high heat until reduced to 1 tablespoon, about 15 minutes. Strain the sauce into another small saucepan. Stir in the crème fraîche and bring to a simmer over moderate heat. Remove from the heat and stir in the softened butter, 1 tablespoon at a time. Stir in the caviar and season lightly with salt. Cover the sauce and keep it warm.
Wipe out the skillet and heat 1/4 inch of oil in it until shimmering. Season the scallops with salt and add half of them to the skillet. Cook over high heat until richly browned, about 1 1/2 minutes per side. Transfer to a plate and keep warm while you cook the remaining scallops.
Rewarm the potato pancakes in the oven. Arrange the pancakes on a platter and set a scallop on each one. Add the chives to the caviar butter sauce and spoon over the scallops. Serve right away.
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