Place a rack about 4 inches from the broiler element and preheat.
Put the potato in a medium saucepan, cover with water by 2 inches, and season generously with salt. Bring to a boil over high heat, lower the heat, and simmer until very tender, about 10 minutes. Drain and cool slightly.
Meanwhile, preheat a large skillet over medium-high heat. Add 2 tablespoons of the butter and heat. Season the salmon all over with salt and pepper. When the butter's foaming subsides, place the salmon, in the skillet and cook, turning once, until well-browned, about 1 1/2 minutes per side. Transfer to a baking sheet.
Put the cooked potatoes in a small bowl with the 2 teaspoons salt, 2 tablespoons butter, half-and-half, and horseradish and, using a fork, mash until fluffy. Season with pepper, to taste.
Melt the remaining 2 tablespoons butter in a small saucepan. Using your hands, press about 1/4 of the potato mixture onto the top of each salmon fillet, to form a thin crust. Lightly brush the potato with the melted butter.
Place the salmon under the broiler and cook until still slightly pinkish inside and the crusts are golden brown, about 10 minutes. Transfer to serving platter and garnish with the chive segments. Serve immediately.
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