Preheat oven to 400 degrees and grease an 8-inch square baking dish with the butter.

In a saucepan, bring the garlic and cream to a simmer. Season with salt and white pepper. Simmer the cream for 15 minutes, or until the cream has reduced by 1/4 and has slightly thickened.

Season the potatoes with salt and white pepper. Alternate layering the white potatoes, sweet potatoes, red potatoes, and 2 cups of the cheese in the prepared dish. You should have a total of six potato layers. Strain the garlic from the cream and slowly pour over the potatoes. Cover the dish tightly with aluminum foil and set on a rimmed baking sheet.

Bake for 1 hour 15 minutes. Check the potatoes for doneness by inserting a thin knife in the center.  The knife should go in smoothly.  If not, return the potatoes to the oven and continue to cook up to 15 minutes longer.  Once the potatoes are done, remove the foil, sprinkle with the remaining 1/2 cup Parmesan, and continue cooking for 15 minutes or until the top is golden brown. Allow the gratin to cool at least 15 minutes before serving.

Garnish with the chopped chives.