Heat oven to 375°F. Melt the butter; brush about 1 tbsp into a 9x13-inch baking dish.
Warm cream, milk, garlic, shallot, thyme, salt, pepper, and nutmeg in a saucepan over medium heat. Simmer gently 3–4 minutes; do not boil.
Slice potatoes 1/8-inch thick with a mandoline if possible. Pat dry to remove excess moisture.
Shingle half the potatoes in the dish. Stir the hot cream mixture and ladle about half over the potatoes.
Sprinkle half the Gruyère and a little Parmesan over the first layer.
Repeat with remaining potatoes, cream mixture, and cheeses to form a second layer.
For a crunchy top, toss breadcrumbs with 1 tbsp melted butter and scatter over the surface. Drizzle remaining 1 tbsp butter over the top (skip if using all the butter for greasing and crumbs).
Cover tightly with foil and bake 40 minutes.
Uncover and bake 25–30 minutes until potatoes are tender and the top is deeply golden and crisp.
Rest 10–15 minutes to set. Broil 1–2 minutes if you want extra color. Slice and serve warm.