Heat the oven and prep the dish: Heat the oven to 375°F and butter a 9x13-inch baking dish. Peel the potatoes and slice them 1/8 inch thick, using a mandoline if you have one. Keep the slices in a bowl of cold water if you're working slowly so they don't brown.

Start the sauce: Melt the 4 tablespoons butter in a large saucepan over medium heat. Add the onion and cook for 3-4 minutes until soft and translucent, then stir in the garlic and cook for 30 seconds until fragrant.

Cook the roux: Sprinkle the flour over the butter and onion and stir constantly for 1 full minute. The mixture will look like wet sand and smell slightly toasty. That minute cooks out the raw flour taste.

Whisk in the milk and cream: Slowly pour in the milk and heavy cream, whisking the whole time so no lumps form. Add the salt, pepper, and nutmeg. Bring to a gentle simmer and cook for 3-4 minutes, whisking often, until the sauce thickens enough to coat the back of a spoon.

Melt in the cheese: Pull the pan off the heat and stir in 1 1/2 cups of the cheddar and 1/2 cup of the Gruyere until completely melted and smooth. Taking it off the burner first keeps the cheese from getting stringy or grainy.

Layer the au gratin potatoes: Drain and pat the potato slices dry. Shingle half of them across the bottom of the baking dish, overlapping the edges. Pour half the cheese sauce over the top and spread it to the corners. Repeat with the remaining potatoes and sauce, then scatter the last 1/2 cup cheddar and 1/2 cup Gruyere over the top.

Bake covered, then uncovered: Cover the dish tightly with foil and bake at 375°F for 40 minutes. Remove the foil and bake for another 30 minutes, until the top is deep golden and bubbling and a paring knife slides through the center with no resistance.

Rest and serve: Let the dish sit for 10 minutes before serving so the sauce sets up instead of running across the plate. Sprinkle with chopped chives right before it goes to the table.