Use a mandoline to slice the potatoes really thin. I use the 1.5 or 2.5mm setting on my handheld mandoline (or 1/16 to 1/8 of an inch).
The potatoes themselves generally release enough oil from the butter and cheese to not require any greasing of a non-stick muffin tin. If you know your pan likes to stick, I'd recommend lightly greasing it.
*Look for cheese specifically labeled vegetarian, if needed.
Press the cheese down a bit when measuring, or have "heaping" measurements. :)
The recipe card was updated 11/2020; the cooking time was reduced by 5-10 minutes after further testing and feedback. The cook time will depend on the thickness of your potato slices and how high you stack them, as noted. Please keep this in mind and check for doneness as the tops and sides become golden brown and bubbly.