Slice potatoes lengthwise on a mandoline, about 1/8 inch thick. Do not rinse starch off.
Melt 1 1/2 sticks of the butter in a small saucepan. Set aside.
Melt 2 tablespoons of butter in each of 4 individual cast-iron skillets. Divide the sliced potatoes into 4 individual portions. Place a third of the potato portions into each skillet. Pour some melted butter over the potatoes and sprinkle each with a pinch of salt and a few grinds of black pepper. Continue layering with potatoes and melted butter and sprinkling with salt and pepper until all the potatoes are used, drizzling the remaining melted butter over the top.
Bake until golden brown, 6 to 8 minutes. Garnish each skillet with a dollop of sour cream and a sprinkle of chives.
This can also be made in an 8-inch cast-iron skillet. Allow for a few extra minutes cooking time.