Turn your Instant Pot onto Saute. Add the olive oil, carrots, onion, celery, garlic, thyme, salt, and pepper. Saute for 2-5 minutes.
Add 4 cups of bone broth, 2 cups of bone broth or water, and the wild rice. Stir to combine everything and make sure nothing is sticking to the bottom of the Instant Pot.
Close the Instant Pot and program it for pressure cooking on high for 20 minutes. You can do the natural release or intermittent release option.
Once all the pressure has been released, open the lid. If the soup seems too thick, add 1-2 more cups of warm bone broth. Stir in the milk and cheese and let the soup simmer for 10 minutes before serving.
You can also turn the Instant Pot onto slow cooker mode after the soup has been pressure cooked if your model has that option. Alternatively, transfer the soup to a pot and let it simmer over the stovetop until you are ready to serve it.