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 degrees. 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 creme fraiche 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.

The pancakes and caviar sauce can stand at room temperature for up to 2 hours. Rewarm the caviar sauce over low heat, stirring constantly.