Preheat a waffle iron to medium-high. Preheat the oven to 200 degrees F. Peel the potatoes, then shred on a box grater. Squeeze out the excess liquid into a bowl and reserve.
Combine the shredded potatoes, butter, chives, 2 teaspoons of the potato liquid, 1/2 teaspoon salt and a few grinds of pepper.
Lightly brush the top and bottom of the waffle iron with melted butter. Pour a scant 1/2 cup of the potato batter into the waffle iron, close gently and cook until golden and crisp, 4 to 8 minutes. Keep the cooked waffles warm in the oven or covered with foil on a plate while you make the remaining waffles.
Top the waffled hash browns with smoked salmon, dollop with sour cream and sprinkle with chives.