Set the oven and clarify butter. Preheat the oven to 375°F and adjust a rack to the center.Clarify the butter: Add the butter to a small sauté pan and turn the heat to low. Melt the butter completely. Use a spoon to remove the milk solids from the top surface. Then, very slowly, pour the butter into a large mixing bowl, leaving any milk solids that have settled to the bottom, in the pan.
Prepare the muffin pan. Use a pastry brush and a small amount of the melted butter to grease 8 cups in a muffin pan. Then line them with parchment paper rounds. (You can get the size right by outlining the bottom of the cups from the muffin tin on the parchment.) Add the thyme and salt to the clarified butter and mix. Set aside.
Slice potatoes. Preferably using a mandoline, slice the potatoes as thinly as possible, but not paper thin! Add all of the slices to the bowl with the butter, and use your hands to be sure the slices separate and all become well coated.
Assemble in pan. Divide potato slices evenly among the prepared muffin cups, layering them in a circular pattern. Lightly press the top of each one to make sure the potatoes are packed. Drizzle any remaining butter-thyme mixture over them.
Bake. Cover the muffin tin lightly with foil and place in the preheated 375°F oven for 20 minutes. The potatoes should now be tender. Remove the muffin tin from the oven, turn it up to 425°F, and cover a baking sheet with parchment paper. Very gently, use a small off-set metal spatula to remove each potato cake from the muffin pan, and place them on the parchment-lined baking sheet. Be sure to take each parchment round off the bottoms as you go. Place them back in the now 425°F oven until the edges of the potatoes are golden, about 10 minutes.