Add the olive oil and 1 tablespoon of butter to your Instant Pot, then set the "saute" function to 6 minutes. Allow 1 minute for the butter to melt, then add the onion, carrots, celery, garlic, 6 oz. of the mushrooms, red pepper flakes, and salt and sauté until the veggies are soft and the timer beeps.
Next, pour in the rice and 32 oz. of the broth and stir. Place the lid on, ensuring the valve on top is in the "seal" position. Pressure cook the soup for 15 minutes, then natural release for 5 minutes before switching the valve on top to "vent" to manually release the remaining pressure.
While the soup is cooking, melt the remaining tablespoon of butter in a small skillet over medium heat, then add the rest of the mushrooms and thyme. Sauté until the mushrooms are slightly golden and crispy, about 5 minutes; set aside.
Remove the lid from your pressure cooker, then stir in the remaining broth, along with the milk and cheese. Taste, then add more salt, pepper and red pepper flakes as desired.
Divide soup into bowls, then top them with the crispy mushrooms and a sprinkle of parmesan. Enjoy!