Preheat the oven according to your rice package directions, generally around 375°F.
Melt the butter with the olive oil in an oven-safe skillet over medium heat.
Add the thinly sliced onions and cook until softened and lightly golden.
Pour in the balsamic vinegar and stir, scraping any flavorful browned bits from the skillet
Add the beef broth and onion soup mix. Whisk until the soup mix dissolves, then bring the mixture to a gentle simmer.
Turn off the heat. Add the rice and spread it evenly throughout the skillet.
Place small pieces of butter over the rice. Cover the skillet with its lid and bake until the rice is tender and has absorbed the broth
Sprinkle the shredded cheese over the cooked rice. Broil until the cheese is melted, bubbly, and golden
Garnish with fresh thyme and serve warm.