Preheat the oven to 375˚. Butter a 9-by-13-inch baking dish.

Heat the olive oil and 2 tablespoons butter in a large skillet over medium-low heat.

Add the onions, 1 teaspoon salt, a few grinds of pepper, and the sugar. Stir, cover, and cook for 10 minutes.

Uncover, increase heat to medium, and cook, stirring for an additional 10 minutes.

Add the thyme, sherry, and broth. Cook until liquid evaporates and onions are caramelized, about 15 minutes.

Melt the remaining 2 tablespoons butter in a dutch oven over medium heat. Stir in the flour and cook for 1 minute.

Whisk in the heavy cream and milk. Add the remaining ½ teaspoon salt and nutmeg. Bring to a simmer, whisking until smooth.

Cook, stirring, until thickened, about 2 to 3 minutes. Remove from heat. Stir in all but ½ cup each of the gruyère and Swiss until melted.

Stir in the sliced potatoes.

Layer half of the potatoes in the baking dish, then half of the onions. Repeat with remaining potatoes and onions.

Sprinkle with reserved gruyère, Swiss, and parmesan. Cover with foil, tenting it to avoid touching cheese.

Bake until potatoes are tender, 45 to 55 minutes. Uncover and bake until golden brown, 20 to 25 minutes.

Let sit for 20 minutes before serving. Top with chives.