Preheat the oven to 180°C (350°F).
Wash and peel the potatoes.
Slice the potatoes finely on a mandolin (about 1 mm thick).
Butter the muffin form.
Grate the parmesan finely.
Place a tbsp of grated cheese on the bottom of each muffin hole.
Arrange the biggest potato slices around the muffin hole's perimeter, ensuring the pieces overlap and reach the bottom.
Place sliced potatoes between the previous ones for the next layer and continue until you have made a potato rose. Use the smallest slices for the center.
Press down gently on the center of each rose before baking to help compress the layers and create a tighter shape.
Melt the butter.
Mix half of the melted butter with the remaining grated parmesan, sea salt, and pepper until it forms a paste.
Spread the paste on the top of the roses to hold the roses together.
Spoon the remaining melted butter over the roses.
Bake in the bottom of the oven for around 40 minutes until golden and bubbly. Rotate the muffin tin halfway through for even browning on all sides.
When they are out of the oven, sprinkle the top of the roses with more sea salt.
Allow to cool slightly.
Use a fork to lift the roses out of the tin carefully.
Sprinkle with fresh thyme or spring (green) onion and serve.