Heat the oven to 400°F (200°C). Grease a 12-cup muffin tin thoroughly with butter or nonstick spray, paying special attention to the sides and bottom of each cup.

Scrub the Yukon Gold potatoes and slice them into very thin, even rounds about 1/16 to 1/8 inch thick. A mandoline gives the most consistent slices, but a sharp knife also works when used carefully.

In a large bowl, combine the melted butter, olive oil, garlic, Parmesan, thyme, kosher salt, and black pepper. Stir until the seasonings are evenly distributed through the buttery mixture.

Add the potato slices and toss carefully until every slice is lightly coated. Keep the slices intact as much as possible so they can form neat, distinct layers.

Stack the coated potato slices in the muffin cups, filling each one slightly above the rim because the potatoes shrink during cooking. Keep the stacks level and pack them firmly without compressing them excessively.

Cover the muffin tin loosely with foil and bake for 25–30 minutes. The covered portion allows the centers to soften before the exposed layers begin to brown deeply.

Remove the foil and continue baking for 20–25 minutes, until the edges are deeply golden and crisp and the centers are tender when pierced with a knife. Watch the final minutes closely because oven temperatures and muffin tins can vary.

Let the potato stacks rest for 5–10 minutes before removing them. Run a thin knife around the edges if needed, then finish with additional Parmesan and optional chives or parsley.