Preheat the oven to 190C (375F). Butter the baking dish so the first potatoes slide and the edges brown evenly.
Thinly slice potatoes 2-3 mm (1/16-1/8 inch) using a mandoline for consistent cooking; you want tender sheets that still hold shape.
In a saucepan, sweat minced garlic in butter over medium heat until fragrant, about 30 seconds, then add cream and milk.
Bring the cream to a gentle simmer - small bubbles around the edge - then whisk in salt, pepper and nutmeg. Do not boil; simmering concentrates flavor without breaking the cream.
Remove the pan from heat and stir in half the grated cheeses until mostly melted, creating a glossy, seasoned cream.
Arrange a single layer of potato slices in the dish, slightly overlapping, then spoon a third of the cream mixture over and sprinkle a touch of cheese. The cream should just cover the potatoes, creating steam and richness.
Repeat two more layers, finishing with the remaining cheese on top for a deeply browned crust that crackles when cut.
Cover loosely with foil and bake for 35 minutes, then remove foil and bake another 20-30 minutes until the top is golden and bubbling and a knife slides through the center easily.
Let rest 10-15 minutes; the gratin will set and the textures will meld. You should smell butter, toasted cheese and a subtle nutmeg aroma.