Wash and peel the potatoes, then slice them thinly, about 1/8 inch thick. Place the slices in cold water to prevent browning.
In a large mixing bowl, combine heavy cream, minced garlic, salt, black pepper, dried thyme, and paprika. Whisk until well blended.
Layer half of the sliced potatoes in a greased baking dish, pour half of the cheese mixture over, and sprinkle half of the cheddar and mozzarella cheeses. Repeat with remaining ingredients.
Drizzle melted butter over the top, cover with foil, and bake in a preheated oven at 375°F for 45 minutes. Remove foil and bake for an additional 15-20 minutes until golden brown.
Let the gratin cool for about 10 minutes before serving.