In a large, high-sided skillet, melt the butter over medium heat.

Add the olive oil, onions and a couple generous pinches of salt and cook, stirring often, over medium heat until the onions are amber in color, about 40 minutes. If the onions are beginning to stick to the skillet, you can add a splash of water, as needed.

Season with salt and pepper, to taste.

Sprinkle the flour over the onions and stir until it has been absorbed.

Add the white wine, and, with a wooden spoon, scrape any brown bits from the pan, and cook until the wine has been absorbed, about 1 minute.

Add the beef broth and thyme and cook until the sauce thickens, for 1 to 2 minutes; season with salt and pepper, to taste.

Meanwhile, preheat the broiler.

Transfer the onions to a 7"x11" casserole dish and level them out with the back of a spoon.

Sprinkle the croutons evenly over the onions and layer the cheese over the croutons, covering the entire top.

Broil until the cheese melts and starts to crisp, about 3 minutes; serve immediately.