In a large skillet over low heat, combine 2 tablespoons of olive oil and 1 tablespoon of butter. Add 2 sliced yellow onions and cook gently for 30-45 minutes, stirring occasionally, until golden brown and soft. Stir in 1 tablespoon of brown sugar and 1 tablespoon of balsamic vinegar.

Using the same skillet after removing onions, add another tablespoon of olive oil and bring to medium heat. Sauté 8 ounces of sliced cremini mushrooms for about 8-10 minutes until browned. Add 2 minced garlic cloves and thyme, splash of white wine if desired; cook until evaporated, around 2 minutes.

Spread about 1 tablespoon of softened butter on one side of two slices of sourdough bread. Spread Dijon mustard on unbuttered side. Layer 1 slice of Swiss cheese, caramelized onions, sautéed mushrooms, and top with another slice of cheese.

Heat the skillet on medium-low, place the sandwiches unbuttered side down, cook for 3-5 minutes until golden brown. Flip, reapply butter, cover skillet, and cook for another 3-5 minutes to melt cheese.

Remove from skillet, let rest for a minute, then cut sandwiches in half and serve immediately.