Peel and thinly slice two yellow onions and set them aside. In a large skillet, melt 2 tablespoons of olive oil and 2 tablespoons of butter over medium-low heat. Once melted, add the onions to the skillet.

Cook the onions on low heat for 30-45 minutes, stirring occasionally until they become deeply golden brown and sweet.

Stir in 1 tablespoon of brown sugar and 1 tablespoon of balsamic vinegar into the onion mixture. Cook for an additional 2-3 minutes, stirring until the mixture is well combined.

Season the onions with salt and pepper to taste. Remove them from heat and transfer to a bowl, setting aside while you prepare the sautéed mushrooms.

In the same skillet, add another tablespoon of olive oil over medium-high heat. Once shimmering, add 8 ounces of sliced cremini mushrooms. Sauté for 8-10 minutes until they are browned.

Add 2 minced garlic cloves and 1 tablespoon of fresh thyme leaves to the mushrooms, cooking for an additional minute until fragrant. Next, deglaze with ¼ cup of dry white wine.

Once the wine has reduced, season the sautéed mushrooms with salt and pepper to taste. Remove from heat and allow to cool slightly.

Spread softened butter on one side of each slice of sourdough bread. On two slices, spread a thin layer of Dijon mustard. Layer on Swiss cheese, caramelized onions, and sautéed mushrooms. Top with another slice of bread.

Cook the sandwich in the skillet over medium-low heat for 3-5 minutes per side, pressing down gently with a spatula.

Once golden brown, carefully remove the sandwiches from the skillet. Slice in half and serve immediately while warm.