
In a large pot over medium-high heat, heat the olive oil. Add the onions and carrots and cook until they start to soften, about 5 minutes.
Add the mushrooms, thyme, salt, and pepper. Cook until the mushrooms start to brown, about 10 minutes. Add the rice and cook with the mushroom mixture for 1 minute.
Add the broth and balsamic vinegar, stir to combine, and bring the soup to a boil.
Reduce the heat to a simmer and continue cooking for 45 minutes, or until the rice is fully cooked.
Stir in the milk or heavy cream before serving.