Fill a large bowl with water and as you peel each asiago potato, place them in the water until you are ready to slice.
Dry each asiago potato as you remove it from water. Discard water. Using a mandolin, slice potato slices horizontally as thin as possible and place them in a large bowl.
In a small bowl, microwave butter with oil, thyme, salt, and pepper until butter is melted. Stir in grated cheeses. Pour butter mixture over half of the sliced potatoes. Add the other half of the potatoes and the rest of the butter mixture.
With your hands, gently work the potatoes and butter mixture to coat.
Spray a 12 cup standard muffin pan with non-stick cooking spray.
Layer slices of potatoes in each cup dividing equally. Discard any liquid left in the bowl. Sprinkle a little additional cheese over each stack and bake for 40-50 minutes or until browned and crispy and cooked through.
Allow sitting for five minutes to set up, then serve, removing each portion from the muffin pan with a soup spoon. Sprinkle with more thyme to garnish.