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Peel and slice the onions in half, and then into 1/8-inch slices. Sweat the onions in the butter until they have a tiny bit of color.
Peel the potatoes, keeping the peeled ones in cold water so they do not discolor. Cut up the potatoes into evenly sized sticks, about 1/4 by 1/4 by 3 inches (they do not have to be perfect). Do not put the potato sticks back in the water as they will lose too much starch.
Pour the anchovy brine into the heavy cream, stir with a fork and add ground white pepper to taste.
Layer the potatoes, onions and anchovy fillets in the casserole, beginning and ending with the potatoes. Pour the cream mixture over the potatoes until almost to the top. Bake until golden brown on top, 45 minutes. Let rest 5 to 10 minutes before serving. This dish can also be reheated slowly in the oven the next day.