Follow these steps to create a dish that will transport you straight to a cozy family gathering:

1. Heat the olive oil and butter in a large skillet over medium heat until the butter is melted and begins to bubble.
2. Add the sliced onions to the skillet, stirring occasionally. Cook for about 15-20 minutes until they are soft and golden brown.
3. Once the onions are caramelized, add the sliced mushrooms to the skillet. Increase the heat to medium-high and cook for another 8-10 minutes, until the mushrooms are tender and have released their juices.
4. Add the minced garlic and cook for an additional 1-2 minutes, stirring constantly.
5. Stir in the balsamic vinegar, salt, and pepper. If using, sprinkle the fresh thyme leaves over the top.
6. Cook for another 2-3 minutes, allowing the flavors to meld together. Taste and adjust seasoning if necessary.

Remember, patience is key to achieving perfectly caramelized onions. They should be soft and sweet but not burnt, so keep an eye on them as they cook.