Peel and thinly slice the yellow onions from root to tip. Heat 2 tablespoons olive oil and 1 tablespoon butter in a 10 to 12-inch skillet over medium-low heat. Add onions, stirring to coat them evenly. Reduce heat to very low and cook for 30 to 45 minutes, stirring occasionally, until deep golden brown and meltingly soft. Properly caramelized onions should be jam-like in consistency and dark amber colored. If they're browning too quickly or getting crispy edges, lower the heat immediately.

Once onions are caramelized, stir in brown sugar and balsamic vinegar, cooking for 2 to 3 minutes until dissolved. Season with salt and pepper, then remove and set aside.

Wipe mushrooms clean with a damp paper towel and slice them evenly. In the same skillet, heat 2 tablespoons olive oil over medium-high heat. Add mushrooms in a single layer without overcrowding (work in batches if needed) and cook for 8 to 10 minutes, stirring occasionally, until golden brown and tender. The mushrooms will release moisture first, then brown beautifully.

Add minced garlic and fresh thyme to mushrooms, cooking for 1 minute until fragrant but not burnt. If using wine, pour it in and scrape up the browned bits from the pan bottom. Let it reduce for 1 to 2 minutes until nearly evaporated. Season with salt and pepper, then remove from heat.

Butter one side of each bread slice generously (butter should be softened for easy spreading). If using Dijon mustard, spread a thin layer on the unbuttered side of two slices. Heat a clean skillet over medium-low heat.

Place all 4 bread slices buttered-side-down in the skillet. Working quickly, divide the Swiss cheese evenly among all 4 slices (about 1 ounce per slice). On 2 of the cheese-topped slices, pile half the caramelized onions and half the sautéed mushrooms on each. Leave the other 2 slices with just cheese.

Cover the skillet with a lid (if available) to help the cheese melt faster and more evenly. Cook for 3 to 5 minutes total until the bread is golden brown and crispy on the bottom, and the cheese is completely melted and gooey.

Carefully use a spatula to flip the plain cheese slices onto the loaded slices to create two complete sandwiches. Press down gently. Remove from heat, cut each sandwich in half diagonally, and serve immediately while hot.