Heat the olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Add the onion, and sauté for 5 minutes.

Add the brown sugar and salt, lower the heat to medium-low, and cook another 10 minutes, stirring occasionally.

Add the vinegar, and scrape any stuck onion bits from the bottom of pan with a spoon. Simmer for 1 to 2 minutes, until the onion mixture thickens, and season to taste with black pepper.

Cool to lukewarm, or store in an airtight container in the refrigerator until needed, up to 5 days. You should have just shy of ½ cup of cooked onions.

Generously butter one side of each slice of bread; these will be the outsides of your sandwiches.

Arrange one slice, buttered side down, on a plate. Spread thickly with jammy onions (about 2 tablespoons per sandwich; you’ll have extra).

Sprinkle with half the grated cheese. Arrange a second slice of bread on top of the cheese, buttered side facing up. Repeat with the remaining slices to make a second sandwich.

Heat a heavy 12-inch skillet over medium-low heat. Once it’s hot, arrange your sandwiches in the pan, and cook them until crisp and deep golden brown, about 5 minutes per side.

If you are making a larger batch and want to keep them warm and melty, arrange them in a single layer on a baking sheet in a 200-degree oven.

When ready to eat, slice the sandwiches in half. Don’t expect leftovers.