Put potatoes into a large saucepan, and cover with cold water. Bring to a boil; reduce to a simmer, and cook until just tender when pierced with the tip of a knife, 15 to 20 minutes. Drain well. Transfer to a rimmed baking sheet, and let stand until cool enough to handle.
Cut off tops and a thin slice from bottoms of 24 potatoes. Using a small melon baller, scoop out flesh from these potatoes, leaving a 1/4-inch-thick shell. Peel remaining 4 potatoes. Press all flesh through a ricer into a medium bowl.
Preheat oven to 375 degrees. Heat milk and 1 tablespoon butter in a small saucepan over medium heat until butter melts. Stir into riced potato, a little at a time, until creamy. Stir in crème fraîche, 1 teaspoon salt, and the chives; season with pepper.
Transfer to a pastry bag fitted with a 1/2-inch plain round tip (such as Ateco #806). Pipe into shells, mounding about 1/2 inch above top edge.
Bake until tops are golden brown and slightly puffed, about 15 minutes. Brush with melted butter, and bake 15 minutes more. Top each potato with about 1/4 teaspoon caviar, and sprinkle with chives.