Bring a small pot of salted water to a boil. Slice off a little from each end of the potato. Cut potato in sixths lengthwise. Cut the wedges into elongated seven-sided ovals; hold potato wedge steady with fingertips and, with a small paring knife, trim all sharp edges. Work from top to bottom of piece in a quick curving movement, turning the piece slightly between each cut. The finished pieces should be of uniform size and shape. Add potatoes to boiling water. Reduce heat to low, and simmer until tender, about 15 minutes. Drain, and reserve.
Bring stock to a boil in a medium saute pan. Reduce heat to simmer. Season both sides of skate with salt and pepper. Place in pan; cover and cook 2 minutes. Turn fish. Cover once again, and cook until opaque, about 2 minutes more. With a fish spatula, transfer skate to 2 plates. Place 3 pieces of potato on each plate.
While fish is cooking, melt butter in a separate medium saute pan over high heat. Cook until it starts to turn golden, about 1 minute. Remove from heat, swirling pan. This swirling motion combined with the remaining heat in the pan will bring the butter to its desired, almost black color. Add lemon juice, capers, parsley, and a pinch of salt and pepper. Spoon sauce over skate.
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