Preheat your oven to 400°F. Lightly grease a 12-cup muffin tin using cooking spray or a thin coat of olive oil.
In a large bowl, combine the olive oil, paprika, cayenne pepper, minced garlic, fresh thyme leaves, lemon juice, Parmesan cheese, salt, and black pepper. Toss the potato slices in the bowl until they are well coated.
Evenly distribute the seasoned potato slices across the muffin tin cups. Neatly stack the slices so that each cup is filled, keeping in mind that the potatoes will shrink during baking. Drizzle the remaining marinade evenly over the stacks.
Cover the muffin tin with a piece of aluminum foil. Put the pan in the preheated oven and bake for 25 minutes. Then, take off the foil and bake for an additional 25 minutes, or until the edges of the potatoes are golden and crispy.
Remove the tin from the oven and cool the stacks in it for 5 minutes. Run a knife around each cup to release the stacks, then gently lift them out. Transfer the potato stacks to a serving plate.
Garnish with extra Parmesan and fresh thyme leaves when serving.