Melt the butter in a wide skillet over medium heat.

Add the sliced onions and 1/2 teaspoon kosher salt, then stir to coat.

Cook the onions for 20–30 minutes, stirring every few minutes and lowering the heat if they brown too quickly.

Add olive oil or a splash of water if the pan looks dry.

Stir in the optional sugar after about 20 minutes, then continue cooking until the onions are deep golden and sweet-smelling.

Remove the onions from the heat and let them cool slightly.

Preheat the oven to 375°F and grease a 9x13-inch glass casserole dish with butter or cooking spray.

In a bowl or large liquid measuring cup, whisk together the heavy cream, 1 cup Parmesan cheese, remaining 1 teaspoon kosher salt, black pepper, and thyme.

Stir the slightly cooled caramelized onions into the cream mixture.

Slice the potatoes thinly, about 1/8 inch thick.

Spread the sliced potatoes evenly in the prepared baking dish, overlapping slightly as needed.

Pour the cream and onion mixture slowly and evenly over the potatoes.

Gently shake the dish or nudge the potatoes with a spoon so the sauce settles between the layers.

Sprinkle the remaining 2 tablespoons Parmesan cheese over the top.

Cover the dish tightly with foil and bake for 45–55 minutes, until a knife slides through the center with no resistance.

Remove the foil and bake for another 15–20 minutes, until the top is golden, bubbling, and lightly crisp at the edges.

Let the potatoes rest for at least 10 minutes before serving.