Begin by thinly slicing the yellow onions. In a large skillet, heat olive oil and butter over low heat. Add the onions and sauté gently for 30-45 minutes until golden-brown.

Once the onions are caramelized, stir in brown sugar and balsamic vinegar, cooking for an additional 2-3 minutes. Season with salt and pepper to taste.

In the same skillet, add a drizzle of olive oil and increase the heat to medium-high. Slice the cremini mushrooms and sauté for 8-10 minutes until golden brown.

Add minced garlic and thyme leaves to the skillet. Cook for another minute until aromatic, then deglaze with white wine and simmer until reduced.

Spread softened butter on one side of each slice of sourdough bread. On the unbuttered side of two slices, spread mustard for extra flavor.

Layer Swiss cheese on each unbuttered side. Divide caramelized onions and sautéed mushrooms between the two sandwiches and top with another slice of Swiss cheese.

Grill the assembled sandwiches in the skillet over medium-low heat for 3-5 minutes each side until the cheese melts and the bread is crispy.

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