In a large pot over medium-high heat, heat the olive oil. Add the onions; sauté until lightly golden, about 5 minutes.
Add all the mushrooms and thyme; sauté until mushrooms begin to brown, about 8 minutes.
Add balsamic vinegar and stir to combine. Mix in the rice and stir for an additional minute. Pour the broth and bring the soup to a boil.
Reduce heat; simmer covered for 45 minutes, until the rice is fully cooked.
Stir in the milk before serving.
