Step 1: Preheat your oven to 375°F (190°C). Grease a 9x13 inch baking dish generously with softened unsalted butter.
Step 2: In a large saucepan, combine the heavy cream, whole milk, minced garlic, salt, pepper, and freshly grated nutmeg. Heat over medium heat, stirring occasionally, until the mixture is hot and just barely simmering, but do not boil. Remove from heat.
Step 3: Gently stir in 1/2 cup of the shredded Gruyère, 1/2 cup of the grated Parmesan, and all of the shredded sharp cheddar into the hot cream mixture until melted and smooth.
Step 4: Arrange a single layer of thinly sliced potatoes in the bottom of the prepared baking dish, overlapping them slightly. Pour about one-third of the cream and cheese mixture evenly over the potatoes. Sprinkle with a portion of the remaining Gruyère and Parmesan.
Step 5: Repeat the layering process two more times, ending with a final layer of potatoes and pouring the remaining cream sauce over the top. Make sure the potatoes are fully submerged in the sauce.
Step 6: Cover the baking dish tightly with aluminum foil and bake for 45 minutes.
Step 7: Remove the foil, and sprinkle the remaining Gruyère and Parmesan cheese evenly over the top. Return to the oven and bake uncovered for another 30-40 minutes, or until the potatoes are tender (test with a fork) and the top is golden brown and bubbly.
Step 8: Once baked, remove the gratin from the oven and let it rest for at least 10-15 minutes before serving. This allows the sauce to set and makes for easier serving.