Start by preheating your oven to 425F. Use a 3.5 Quart oven-safe braiser, preferably with a lid. If your oven safe skillet has no lid, you can cover it with aluminum foil.
Preheat the braiser on medium heat. Melt down 2 tablespoon butter and add in the olive oil. Add the sliced onions and cook low and slow for 15 minutes, stirring often until they start to gain color.
Deglaze the pan with the beef broth and balsamic vinegar. Whisk in the onion soup mix into the broth. Bring the broth a boil and then turn off the heat.
Sprinkle the rice evenly into the hot liquid. Place the lid on the braiser and transfer to the preheated oven. Bake for 30 minutes.
Remove the lid from the cooked rice. Carefully sprinkle on the grated cheese on top of the French Onion Soup Rice. Place the braiser under your oven broiler, watching until just lightly brown.
Remove the French Onion Soup Rice from the oven and garnish with the fresh thyme leaves. Rest for 5 minutes and serve.