Preheat oven to 375°F (190°C). Generously grease a 9x13-inch ceramic or glass baking dish with 1 tablespoon softened butter.
Wash and scrub the Yukon gold potatoes. Using a mandoline slicer, slice potatoes into uniform 1/8-inch rounds. Do not rinse the sliced potatoes to keep their natural starch.
In a medium saucepan, combine the heavy whipping cream, whole milk, remaining 1 tablespoon butter, minced garlic, fresh thyme leaves, sea salt, cracked black pepper, and nutmeg. Warm over medium-low heat for 4 to 5 minutes until steaming and fragrant (do not boil). Remove from heat.
In a bowl, toss the shredded sharp cheddar and Gruyère cheeses together so they are evenly mixed.
Arrange one-third of the potato slices in a slightly overlapping shingled layer in the prepared baking dish. Pour one-third of the warm garlic cream sauce evenly over the potatoes, then sprinkle with one-third of the cheddar-gruyère cheese blend.
Repeat with a second layer of potatoes, one-third of the cream sauce, and one-third of the cheese blend.
Finish with the final third of potatoes and the remaining cream sauce. Top generously with the remaining cheddar-gruyère blend and scatter the grated Parmigiano-Reggiano over top.
Cover dish tightly with aluminum foil (tenting slightly so cheese does not stick). Bake for 45 minutes. Remove foil and continue baking uncovered for 25 to 30 minutes until potatoes are fork-tender and the cheese crust is bubbling and golden-brown with caramelized spots.
Remove from oven and let rest for 15 minutes before serving to allow the layers to settle and set cleanly. Garnish with fresh thyme sprigs and cracked pepper!