Arrange a rack in the middle of the oven and heat the oven to 375ºF. Coat the wells of a standard 12-well muffin pan with olive oil. Finely grate 2 garlic cloves and place in a large bowl. Add 3 tablespoons olive oil, 1 teaspoon kosher salt, 1/2 teaspoon ground black pepper, and 1/4 teaspoon cayenne pepper, and whisk to combine.

Grate 2 ounces cheese on the small holes of a box grater (about 1 loosely packed cup). Reserve 1/4 cup of the grated cheese for broiling at the end. Peel 1 1/2 pounds russet or Yukon gold potatoes. Using a mandoline or a sharp knife, very thinly slice the potatoes into rounds about 1/16-inch thick. Add to the bowl of garlic oil and toss the potatoes with your hands until every slice is coated in the oil.

Working with the largest slices first, fill each muffin well with 4 stacked slices of potato. Top each stack with a small pinch of grated cheese (about a scant 1/2 teaspoon). Repeat 2 more times. You may not have enough potatoes to do a stack of 4 slices on the last round, so divide the remaining slices evenly among the wells if needed. Sprinkle with any remaining cheese.

Bake until the potatoes are tender and cooked through, about 30 minutes. Remove the pan from the oven. Set the oven to broil on high. Sprinkle the potato stacks with the reserved 1/4 cup cheese. Broil until the cheese on top is melted and browned, and the edges are crispy, about 4 minutes.

Run a small offset spatula or knife around the edge of each muffin well to loosen the potato stacks. Immediately transfer the stacks to a serving dish and sprinkle with 2 tablespoons finely chopped fresh with chives if desired.