Preheat your oven to 375°F. Spray 12 standard muffin cups with nonstick cooking spray.
In a large bowl, whisk together the butter, olive oil, parmesan, chopped thyme, kosher salt and freshly cracked black pepper. Add the potato slices and toss to coat evenly.
Layer the potato slices into stacks in the prepared muffin cups, filling each one to the very top. Bake until the edges and tops are golden brown and the centers are tender, 45 to 55 minutes. Remove the muffin tin from the oven and let the Parmesan Potato Stacks rest for 5 minutes. Carefully remove them with a small spoon and sprinkle with thyme leaves and extra Parmesan cheese if desired along with a sprinkle of salt and pepper. Serve immediately.