Trim each of the sea bass fillets into a 5-by-1 1/2-inch rectangle. Season with salt, pepper and the thyme.
Using a knife, slice each potato lengthwise to remove the rounded portions and make a block shape; do not cut off the tips of the potatoes, but do peel them. Using a mandoline, slice the potatoes lengthwise into very thin slices. Brush the slices on both sides with the melted butter and season with salt.
For each paupiette, on a 10-inch sheet of wax paper, arrange 8 slightly overlapping potato slices to form a 5-inch wide rectangle. Center a fish fillet horizontally in the rectangle and wrap the potatoes over and around the fillet to enclose it completely. Use the wax paper to help seal the potato wrap around the fish; remove the paper. Repeat to form the remaining paupiettes. Cover and refrigerate for up to 2 hours.
In a medium skillet, melt 2 tablespoons of the butter over moderate heat. Add the leeks and cook, stirring, until softened, about 5 minutes. Season with salt and pepper and keep warm.
In a large nonstick skillet, melt the remaining 2 tablespoons of butter over moderately high heat. Add the paupiettes and cook until the potatoes are tender and golden, turning once with a large spatula, 8 to 10 minutes per side.
Spoon the leeks onto 4 warmed serving plates and ladle the Barolo Sauce around them. Set the paupiettes on the leeks and top each one with a thyme sprig. Garnish with the chives and serve.
The leeks and the red wine sauce find their match in a light red. A California Pinot Noir would be perfect.
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