Begin by slicing the large yellow onions thinly. In a skillet, heat a combination of olive oil and butter over medium-low heat. Add the onions and sauté for 30-45 minutes until golden brown. Stir in the brown sugar and balsamic vinegar, cooking for an additional 5 minutes. Season with salt and pepper and set aside.

Using the same skillet, add more olive oil and introduce the cleaned cremini mushrooms. Increase heat to medium and sauté for 8-10 minutes, allowing them to brown. Add minced garlic and fresh thyme, cooking for 1 minute. Deglaze with white wine and stir to incorporate.

Spread softened butter on one side of each sourdough slice. On two slices, spread Dijon mustard on the unbuttered side to ensure a golden crust.

Start assembling the sandwich with a buttered slice of bread, adding Swiss cheese, then caramelized onions, then sautéed mushrooms, and topping with another slice of Swiss cheese and a buttered slice of bread.

Preheat the skillet over medium heat. Grill the assembled sandwich for 3-4 minutes until golden brown, then flip and grill the other side until equally crispy.

Remove from skillet and let cool slightly before slicing in half. Serve warm.