In a large deep dish, heat 1 table spoon of olive oil over medium heat, add the chopped onion, and a pinch of salt. Sauté for 10 minutes, then add the minced garlic and cook until the onions turn golden brown.
Add 4 cups of chicken broth, cover, and bring to a boil over medium heat. Once boiling, turn off the heat and start cooking the rice.
Heat the remaining 2 table spoon of olive oil, add the sliced mushrooms with some salt, and cook until the mushrooms darken.
Remove the dish from the oven, add the remaining 1 cup of chicken broth, cheese, butter, salt, and pepper, and stir for 2 minutes. Add the mushroom mixture and serve garnished with thyme.