In a large skillet or sauté pan, melt the butter and olive oil over medium heat until fully combined.
Add the sliced onions to the pan, stirring well to coat them with the butter and oil mixture.
Sprinkle the brown sugar, salt, black pepper, and dried thyme over the onions, stirring to combine.
Reduce the heat to low and cover the pan. Allow the onions to cook for about 30 minutes, stirring occasionally to prevent sticking.
After 30 minutes, remove the lid and increase the heat to medium. Continue cooking the onions for an additional 15-20 minutes, stirring frequently, until they are golden brown and caramelized.
Once the onions reach a deep golden color, drizzle the balsamic vinegar over them and stir well. Cook for another 2-3 minutes to allow the vinegar to meld with the onions.
Remove the pan from heat and let the onions rest for a few minutes before serving. Enjoy them warm as a side dish.