Preheat oven to 400 degrees F (200 degrees C). Generously grease a 12-cup muffin tin with melted butter, reserving the rest for the potatoes.

Slice potatoes to about 1/16 to 1/8 inch thickness using a mandoline or vegetable slicer. Discard end pieces that are mostly skin. Place slices in a bowl and add rosemary, salt, black pepper, and cayenne pepper. Drizzle in the remaining melted butter. Toss with your hands to evenly distribute, about 2 to 3 minutes, trying to separate all the potato slices. Grate about a quarter of the cheese over the potatoes. Toss. Add more cheese; toss again. Continue until all cheese is incorporated.

Stack potato slices in uneven layers (to resemble flower petals) in prepared muffin tin, ending with some of the smaller pieces.

Bake in preheated oven until potatoes are tender, 45 to 60 minutes, testing with a metal skewer for doneness. Potatoes should be very soft. If potatoes look too brown but aren't yet cooked through, cover pan with foil for the final minutes of baking.

Loosen edges of potato roses with a fork and lift out of pan to serving dish.