Preparation

• In a large, heavy skillet over medium heat, melt the butter. Add the onion and 1/2 tsp. salt, stir to coat, cover, and reduce the heat to medium low. Cook for 10 minutes, stirring once or twice, using the onions to sweep the pan as you stir.
• Uncover the pan, add the broth and thyme, and cook, stirring occasionally, until the onions are very soft and the liquid has evaporated, about 15 minutes. Add the sherry, increase the heat to high, and stir until the alcohol cooks off, about 30 seconds. Season to taste with salt.
• Brush tops of four slices of bread with oil. Place oil side down on a large griddle. Divide three-quarters of the cheese among the bread slices. Divide the onions over the cheese, then sprinkle with the remaining cheese. Top the sandwiches with the remaining bread, and brush the tops with oil.
• Heat the griddle over medium heat. Weight the sandwiches (a baking sheet topped with a few heavy cans works well), and cook until the bottoms are deeply golden and the cheese is melty, about 4 minutes. Flip the sandwiches and cook, moving the sandwiches around the griddle as needed to brown evenly, until the second sides are golden, about 2 minutes more. Cut the sandwiches in half and serve.