Heat 2 tablespoons olive oil in a medium saucepan over medium heat. Sauté onion in hot oil until dark golden brown, 15 to 20 minutes.
Remove from heat and stir in balsamic vinegar; set aside.
Heat remaining 2 tablespoons olive oil in a large skillet over medium heat. Cook and stir rice in hot oil for 2 minutes.
Pour in wine while stirring. Reduce heat to medium-low.
Pour in 1 cup hot broth; cook and stir until broth is absorbed.
While stirring constantly, continue adding remaining broth, 1 cup at a time, until all broth is absorbed and rice is tender yet firm to the bite, about 20 minutes.
Stir in reserved onion mixture until heated through. Remove from heat.
Stir in chicken and butter; season with salt and pepper. Garnish each serving with thyme.
