In a large skillet, melt the butter and olive oil over medium heat until bubbling.
Add the sliced sweet onions to the skillet, stirring to coat them in the butter and oil.
Sprinkle the salt and black pepper over the onions, stirring well to combine.
Cook the onions, stirring occasionally, for about 15-20 minutes, until they become soft and translucent.
Add the brown sugar and balsamic vinegar to the skillet, stirring to coat the onions evenly.
Continue to cook for an additional 10-15 minutes, allowing the onions to caramelize and develop a rich color.
Once the onions are deeply caramelized, stir in the fresh thyme leaves and cook for another minute.
Remove the skillet from heat and serve the Tennessee onions warm as a side dish or topping.