Bring a stockpot of lightly salted water to a boil over high heat. Add the potatoes, and cook until tender, 15 to 20 minutes. Drain potatoes. Cut potatoes into 1/3-inch-thick rounds, and transfer to a bowl. Set aside. Bring a stockpot of lightly salted water to a boil over high heat, and prepare an ice bath. Trim bottoms from Brussels sprouts. Peel away the inner leaves, one by one. Add carrots to boiling water, and cook until tender, about 4 minutes. Using a slotted spoon, transfer carrots to bowl with the potatoes. Add the leaves of the Brussels sprouts to the boiling water, and cook until just tender, about 3 minutes. Using a slotted spoon, transfer leaves to the ice bath to stop the cooking process. Drain in a colander, and add leaves to bowl with carrots and potatoes. In a medium saucepan, bring 1 cup water to a boil. Add the cream, and return to a boil. Boil 2 minutes, reduce heat to low, and whisk in the butter, piece by piece. Add the reserved potatoes, carrots, and Brussels sprouts. Season with salt and pepper. Cook until the vegetables are heated through, and keep the vegetable ragout warm. In a 12-inch nonstick saute pan, heat the oil over high heat until very hot but not smoking. Season the scallops with salt and pepper. Cook scallops until golden brown on bottoms, about 3 minutes. Using kitchen tongs, turn scallops; reduce heat to medium. Cook scallops until the other sides have browned, but are still rare inside, about 2 minutes. Reserving 1 cup vegetable ragout, divide equal amounts of the remaining ragout among 4 warm dinner plates. Arrange 3 scallops on top, then top each with 1/4 cup reserved ragout. Spoon the Scallop-Butter Sauce around the plates, and sprinkle with chives. Garnish each dish with equal amounts of caviar, and serve immediately.
Combine the scallop scraps, 1/4 cup water, and the white wine in a small nonreactive saucepan. Set the pan over medium-high heat, and bring to a boil. Cook until the liquid has reduced by half, about 7 minutes. Add cream; boil until liquid has reduced by half, about 10 minutes. Remove pan from heat, and whisk in the butter, piece by piece. Season with salt and pepper. Serve warm.
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