Move an oven rack to the middle position and preheat the oven to 375°F (190°C). Grease the bottom and sides of an 8x8-inch or 9x9-inch baking dish with 1 tablespoon of the butter and set it on a rimmed baking sheet.
Peel the potatoes and slice them 1/8 inch (3 mm) thick on a mandoline. Keep the slices together in a large bowl and do not rinse them.
Shred the Gruyère and cheddar. Combine 1 cup Gruyère and 3/4 cup cheddar for the sauce, and the remaining 1/2 cup Gruyère and 1/4 cup cheddar with the Parmigiano-Reggiano and panko for the topping.
Melt 3 tablespoons butter in a heavy 3-quart saucepan over medium heat. Cook the onion for 4 to 5 minutes until soft and translucent, then stir in the garlic and cook for 30 seconds.
Whisk in the flour for 1 to 2 minutes, then slowly pour in the milk while whisking. Whisk in the cream and bring to a gentle bubble, then simmer for 3 to 4 minutes until thickened. Off the heat, stir in the salt, pepper, nutmeg, thyme, and sauce cheese until smooth.
Layer one-third of the potatoes in the dish, top with one-third of the sauce, and repeat twice more. Press down gently with a spatula so the layers sit flat.
Cover the dish tightly with foil and bake for 45 minutes, until the sauce is bubbling around the edges.
Toss the melted butter with the topping mixture and scatter it evenly over the surface. Bake uncovered for 25 to 30 minutes, until deeply golden and a knife slides through the potatoes easily.
Let the dish rest for 15 minutes, sprinkle with parsley, and cut into 8 portions.