Pat chicken breasts completely dry with paper towels. Season generously with 1 tsp salt and ½ tsp black pepper. Allow to sit at room temperature for 15-20 minutes while preparing the onions.

Cut onions in half from pole to pole, then slice into thin half-moons approximately ⅛-inch thick.

Heat a 12-inch skillet over medium-low heat. Add 2 tbsp butter and 1 tbsp olive oil. Once butter is melted, add all sliced onions and ½ tsp salt. Stir to coat evenly.

Cook onions for 25-30 minutes, stirring every 3-4 minutes. They will gradually reduce in volume and develop a deep golden-brown color. Add optional brown sugar at the 15-minute mark if caramelization is slow.

When onions reach a deep amber color, add minced garlic and cook for 60 seconds. Add balsamic vinegar and deglaze the pan, scraping up browned bits. Cook until vinegar is mostly reduced, about 1-2 minutes. Transfer caramelized onion mixture to a bowl.

Increase heat to medium-high. Add remaining 1 tbsp each of butter and olive oil to the skillet. Once shimmering, place chicken breasts presentation-side down. Sear undisturbed for 5-6 minutes until golden brown.

Turn chicken and cook for 4 minutes on the second side until internal temperature reaches approximately 155°F.

Return caramelized onions to the pan around (not on top of) the chicken. Add chicken stock, thyme, and Dijon mustard, stirring to incorporate.

Reduce heat to medium-low, cover partially, and cook for 3-4 additional minutes until chicken reaches 165°F internal temperature at the thickest point.

Remove skillet from heat. Transfer chicken to a cutting board and let rest for 5 minutes. Meanwhile, stir remaining tablespoon of butter into the onion mixture. Slice chicken against the grain and serve topped with caramelized onions.