Thinly slice the yellow onions and heat a skillet over medium-low heat. Add olive oil and butter, cooking the onions slowly for 30-45 minutes until golden brown. Stir in brown sugar and balsamic vinegar, cooking for an additional 5 minutes until syrupy.

In the same skillet, increase heat and add more olive oil if needed. Sauté cremini mushrooms for 8-10 minutes until browned and tender. Add minced garlic and thyme, cooking for another minute. Optionally de-glaze with white wine and season with salt and black pepper.

Spread softened butter on one side of each slice of sourdough bread. On the unbuttered sides of two slices, optionally spread Dijon mustard.

Assemble the sandwich by placing one slice of bread butter side down on a surface. Layer with Swiss cheese, caramelized onions, and sautéed mushrooms. Top with another slice of Swiss cheese and close with the second slice of bread, butter side facing out.

Grill the sandwich in a skillet over medium heat for 3-4 minutes on one side, then flip and grill for another 3-4 minutes until toasted and cheese is melted.

Remove from skillet, allow to cool slightly, then slice and serve warm, ideally with a side salad or tomato soup.