Heat the oven to 425°F with a rack in the upper third. Set a 10-inch oven-safe skillet on the stove so it is ready when you need it.

Put the sourdough slices on a small sheet pan.

Toast the bread in the oven for 5 minutes, until the slices feel dry at the edges and just lightly crisp. Take the sheet pan out and set it aside.

Set the skillet over medium heat. Add the olive oil and butter to the skillet.

When the butter has melted, add the sliced onions and 1/2 teaspoon salt to the skillet. Stir well so the onions are coated with the fat.

Cook the onions for 5 minutes, stirring often, until they soften and shrink.

Turn the heat down to medium-low. Cook the onions for 15 to 18 minutes more, stirring every few minutes, until they are very soft and deep golden. If the skillet starts to brown too fast, add 1 tablespoon water to slow it down.

Stir the dried thyme and black pepper into the skillet. Cook for 30 seconds, just until the thyme smells fragrant.

Pour the vegetable broth into the skillet. Stir and scrape the bottom of the skillet with a wooden spoon to loosen any browned onion bits.

Let the broth simmer for 3 to 4 minutes, until the liquid reduces a little and the onions look glossy and lightly saucy instead of soupy.

Sprinkle 3/4 cup of the shredded Gruyere over the toasted bread on the sheet pan. Leave the remaining 1/4 cup Gruyere for the skillet.

Use the back of a spoon to make 6 shallow spaces in the onion mixture. Sprinkle the remaining 1/4 cup Gruyere over the onions in the skillet.