Soak the cannellini beans for 15 minutes in cool water while you prep the rest of the ingredients.
Lightly grease the inner pot of the Instant Pot. This is optional but helps prevent the rice from sticking to the bottom and makes cleanup easier. If you are oil-free, skip this step.
Place the onions, carrots, celery, garlic, mushrooms, soaked cannellini beans, wild rice, vegetable broth, water, 2 teaspoons kosher salt, black pepper to taste, and bouquet garni in the Instant Pot. Stir to combine.
Secure the lid and set the Pressure Release to Sealing. Select the Pressure Cook (or Manual) setting at high pressure and set the cook time to 45 minutes.
Once the 45-minute timer has completed and beeps, perform a quick pressure release by carefully switching the Pressure Release knob from Sealing to Venting. Open the pot and remove the bouquet garni.
While the pressure is releasing, make the slurry: whisk together the plant-based milk and cornstarch or arrowroot powder, whisking well until no clumps remain.
Select the Sauté setting. Once the soup is starting to boil, add the cornstarch or arrowroot slurry to the Instant Pot, along with a few pinches of kosher salt and stir well to incorporate. Heat for approximately 5 minutes, or until the soup has thickened (arrowroot powder takes less time to thicken). The soup will also thicken as it rests.
If you want the soup to be extra creamy, run an immersion blender through about half of the soup (or transfer half of it to a stand blender) and roughly purée. Stir well so the unblended soup is incorporated into the blended soup.
Store leftovers in an airtight container in the fridge for up to 5 days.