Preheat air fryer to 330° F. Using a pastry brush, lightly brush 4 foil pieces, muffin tin cups, or ramekins with oil.
Using a Mandoline, thinly slice the potatoes, about 2 mm (or 1/16 of an inch) thick. Starting with the smaller pieces, lay a potato slice on the oiled foil, and brush each potato slice with a thin layer of oil. Lay another slice of potato on top, and brush with oil. At every third layer, sprinkle with a little salt and pepper. Repeat stacking, brushing with oil, and seasoning until you have used up the potato slices. At the top of each stack, season with salt and pepper, and sprinkle with fresh thyme leaves. Loosely wrap the foil around the outside of the potato stack.
Air fry at 330° F for about 35-40 minutes or until the potatoes are golden brown and cooked through. Remove the potato stacks from the air fryer basket, leaving them in their foil containers. Garnish with additional fresh herb leaves as desired.
When the potatoes are finished cooking, the stacks might be a little unstable. If you have any that slid sideways, this is the best time to straighten them.
After straightening, let the stacks cool for about 5 minutes in the foil. This will let them stick together better and you should then be able to easily remove the entire stack by peeling away the foil.