Preheat the oven to 350˚F. Lightly oil a muffin tin.
Peel and thinly slice the potatoes. I use a hand held mandolin, and put it on the second thickest setting, a little less than 1/8" thick.
Spread 1 teaspoon salsa on the bottom of each muffin hole. This, in addition to the oil, keeps the potatoes from sticking to the bottom.
Take two potato slices and place them in each muffin hole on top of the salsa. Lightly brush with butter, sprinkle with a pinch of the grated cheese (about 3/4 teaspoon), and a pinch of salt and pepper. Stack two more potato slices on top, the same size or a little smaller than the bottom slices. Repeat with the butter, cheese, salt and pepper. Add two more potato slices the same size or a smaller, and top with one pickled jalapeño slice. Finish with one potato slice, butter, salt and pepper, and divide the black beans between the muffin holes. Be sure to retain enough grated cheese to top the stacks at the end.
Cover with foil and bake for 15 minutes. Remove the foil, divide the remaining cheese between the potato stacks, and broil for 2 minutes. The potatoes should be a little crispy around the edges.
Scoop the potato stacks from the muffin tin onto a serving board using a large spoon.
Top each stack with avocado and a dollop of sour cream.
Serve warm with napkins.