In a large skillet over medium-high heat, melt two tablespoons of the butter. Add the onions and season with salt and pepper. Cover and cook, stirring occasionally until the onions are softened, three to five minutes.

Uncover, reduce the heat to medium-low, and cook, stirring occasionally, until deeply red and darkened, 20 to 25 minutes. If the onions look dry, add a few tablespoons of water to the skillet, scraping up any browned bits that are stuck to the bottom.

Deglaze the skillet with vinegar and cook until the liquid has evaporated, about one minute. Transfer the onions to a medium bowl and season to taste with salt and pepper. Wipe out the skillet.

Divide the apricot jam evenly between each slice of sourdough. Divide the cheese evenly between two of the slices, then top each slice with half of the onion mixture. Top with fresh basil leaves, then top with the remaining slices of bread and press down gently.

In the skillet, melt one tablespoon butter over medium heat. Add the sandwiches and cook until the bottoms turn toasty and golden brown, three to four minutes, reducing the heat to prevent toast from darkening too quickly, if needed. Add the remaining one tablespoon butter, flip the sandwiches, press down and cook until the cheese has fully melted and the bottoms turn golden brown, three to four minutes. Serve immediately.