Thinly slice the onions, heat a skillet over low heat, and add olive oil and butter. Once melted, add onions and caramelize for 30-45 minutes, stirring occasionally. Stir in brown sugar and balsamic vinegar in the last few minutes.

Slice the cremini mushrooms. In another skillet, heat olive oil over medium-high heat. Add mushrooms and sauté for 5-7 minutes until golden brown. Add garlic and fresh thyme, sauté for an additional minute.

Deglaze the skillet with dry white wine, scraping the browned bits. Cook for 2-3 minutes, then season with salt and pepper.

Butter one side of each slice of bread, spread Dijon mustard on the unbuttered side of two slices. Layer with Swiss cheese, caramelized onions, and sautéed mushrooms. Top with remaining slices, buttered side facing out.

In a skillet over medium heat, cook sandwiches for 3-5 minutes per side. Cover with a lid to help melt the cheese.

Once golden and melty, slice sandwiches in half and serve warm with sides like salad or tomato soup.