Mix together the potatoes, onion, Parmesan, nutmeg, salt and pepper. Generously butter 6 muffin tins, 2-1/2" size. Press the potato mixture evenly into the muffin cups. Bake 45 minutes or until the edges are well browned.
Remove from the oven to a rack to cool slightly. Loosen and invert the potato cups onto a plate.
Prepare Marinated Cucumber Slices.
While the potatoes are baking, notch the edges of the cucumber with the tines of a fork. Using a mandoline slice approximately 7 to 8 slices per person.
Mix the mint, vinegar, sugar and cayenne together and marinate the cucumber slices for approximately 30 minutes.
To Plate Up: Overlap the marinated cucumber slices in a circle off to the side on each plate. Place the inverted potato cup on top, add a sprig of Parsley and serve.