In a large skillet, heat olive oil and butter over medium-low heat. Add sliced onions and cook for 30-45 minutes until deep golden brown. Mix in brown sugar and balsamic vinegar for the last 2-3 minutes.

Remove onions and add more olive oil, sauté sliced mushrooms for 8-10 minutes. Stir in minced garlic and fresh thyme, adding white wine if desired.

Butter one side of each slice of sourdough bread. Spread mustard on the unbuttered side (optional), layer Swiss cheese, caramelized onions, and sautéed mushrooms, topping with another slice of cheese and a second piece of bread.

Cook the assembled sandwiches for 3-5 minutes on one side until golden brown. Flip and cover with a lid to melt cheese, grilling the other side for another 3-5 minutes.

Cut sandwiches in half and serve immediately with a side salad or tomato soup.