Preheat a heavy skillet over medium heat. In a mixing bowl, toss the potatoes with half of the melted butter. Divide potato mixture into 18 equal portions, making a small mound with each.
Add the remaining melted butter to the skillet and heat. Carefully drop the mounds into the heated butter, and fry for about 5 minutes on each side, or until crispy. (Keep in mind that slow cooking is better than fast; hot rapid cooking may burn the potato cakes rather than crisp them).
Remove potato cakes onto paper towels to drain.
Preheat the oven to 350 degrees F.
To make the sauce, heat cider vinegar and shallots in a saucepan over medium heat. Reduce the vinegar by 2/3; the finished reduction should be as thick as syrup.
Add the apple cider and reduce by half.
Add the cream, turn the heat down to low, and simmer for 15 minutes.
While cream is simmering, assemble potato cakes as follows: In an ovenproof dish, make 6 stacks, each 3 cakes-high, by starting each stack with a potato cake, topping it with a slice of goat cheese, placing a dollop of applesauce on the cheese, and continuing to layer the cakes, cheese, and applesauce until all the stacks are completed. Then place the dish in the oven for about 8 minutes to reheat.
Just before serving, stir the mustard and 3 tablespoons chives into the sauce, and spoon a little onto each serving plate.
Place a potato cake stack on top of the sauce, and finish each stack with another dollop of applesauce.
