Preheat the oven to 350°F (175°C).
Wash and peel the potatoes well. Using a mandoline slicer, cut the potatoes into approximately 1 mm-thick slices.
Lightly grease each muffin cup with oil or butter. Sprinkle about half a tablespoon of grated Parmesan into each cup.
Arrange the potato slices around the edges of each muffin cup, slightly overlapping them.
Continue layering the potato slices in a spiral, working your way toward the center, until they resemble a rose. Use the smallest slices for the center.
Gently press the center of each potato rose with your finger to keep the layers together and help it hold its shape.
In a small bowl, combine 2 tbsp melted butter with the remaining Parmesan, salt, and black pepper. Spread the mixture evenly over the potato roses. Drizzle the remaining melted butter over the roses.
Place the muffin tray in the oven and bake for about 40 minutes. After about 20 minutes, turn the tray around so the potatoes cook evenly. The potato roses are done when the edges are golden and crispy, and the centers are soft.
Remove the tray from the oven and let the potato roses cool for a few minutes. Season with a pinch of salt and garnish with fresh thyme or chopped spring onions. Serve warm.