Begin by slicing the large yellow onions thinly, aiming for uniform pieces to ensure even cooking. In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium-low heat. Once the butter is melted and starts to foam, add the sliced onions. Stir to coat them well, and let them cook gently for 30-45 minutes, stirring occasionally, until they are deep golden brown and caramelized.

After achieving a beautiful caramel color, sprinkle in 1 tablespoon of brown sugar and drizzle 1 tablespoon of balsamic vinegar over the onions. Stir everything together and cook for an additional 2-3 minutes until the sugar dissolves and melds into the onions, enhancing their sweetness. Season lightly with salt and freshly ground black pepper, then set the caramelized onions aside in a small bowl to keep warm.

In the same skillet, add another tablespoon of olive oil and increase the heat to medium-high. Clean and slice the cremini mushrooms into thick pieces, then add them to the skillet. Sauté for approximately 8-10 minutes, stirring occasionally, until the mushrooms are browned and their moisture has evaporated, giving them a rich, meaty texture. Remove the skillet from heat when they are nicely browned.

Next, add 2 cloves of minced garlic and a teaspoon of fresh thyme leaves to the sautéed mushrooms in the skillet. Continue cooking for about 1 minute until fragrant, stirring continuously to prevent burning. This step enriches the flavor of the mushrooms, making them an essential component of your Swiss Mushroom Melt.

Pour in ¼ cup of dry white wine to deglaze the skillet, scraping the bottom to release any flavorful bits stuck to the pan. Allow the wine to simmer for 1-2 minutes until it reduces slightly, intensifying the flavors of the mushrooms. Once done, season the mushrooms with a pinch of salt and set aside, allowing them to stay warm while you prepare the sandwich assembly.

Take your slices of sourdough bread and butter one side of each slice generously using 1 tablespoon of softened butter. On the unbuttered side of two slices, spread a thin layer of Dijon mustard for a little kick. Layer the Swiss cheese evenly, followed by a generous portion of the caramelized onions, and finally, the sautéed mushrooms to create your Swiss Mushroom Melt Sandwich.

Place the assembled sandwiches into the skillet over medium heat, unbuttered side down. Grill for about 3-5 minutes or until the bread is golden brown and crispy. Carefully flip the sandwiches using a spatula, ensuring the fillings remain intact, and grill the other side for an additional 3-5 minutes, or until the cheese melts beautifully.

Once both sides are golden and the cheese has fully melted, remove the sandwiches from the skillet. Allow them to cool for a moment before cutting each sandwich in half. Serve your delightful Swiss Mushroom Melt Sandwich warm, possibly with a side salad or a comforting bowl of tomato soup for a perfect meal experience.