Heat 2 tbsp olive oil over medium heat in a large skillet. When shimmering, add onion, garlic and mushrooms. Turn heat down to medium-low and saute slowly, stirring frequently, until onions are golden brown, 20-25 minutes. Add balsamic vinegar, stir well and set aside.
Meanwhile, heat remaining oil in a saucepan over medium heat. When shimmering, add rice. Stir until rice turns opaque. Add white wine and simmer until nearly dry. Add chicken broth, 1/2 cup at a time, stirring frequently, waiting until each addition is nearly absorbed before adding the next. Continue adding hot broth until rice is fully cooked, about 40-45 minutes.
Stir onions into rice. When heated through, remove from heat and add butter and chicken. Season with salt and pepper to taste. Garnish with thyme and parmesan cheese before serving.