Heat large skillet (I prefer using coated cast iron) over medium heat and add the olive oil and butter, allowing the oil to warm and the butter to melt.
2 tbsp olive oil, 2 tbsp butter
Add the onions, stirring to coat in the oil and butter.
1 large red onion, 1 large yellow onion
Cook until the onions soften, stirring occasionally.
Reduce heat to medium-low and sprinkle with the salt and sugar.
1/2 tsp salt, 1 tsp sugar
Cook until the onions are light brown and begin to caramelize, about 20 minutes.
Add the balsamic vinegar and toss to coat.
1 tbsp balsamic vinegar
Continue cooking until onions are fully browned and caramelized, about 10 more minutes.
Remove from pan and set aside.