Season chicken with salt and pepper. Place the chicken, broth, wild rice, garlic, onion, carrots, celery, rosemary, oregano, thyme, and mushrooms in the Instant Pot.
Lock the lid into place. Turn the vent valve to ‘Sealing’ position.
Select ‘Manual’ setting on high pressure and set cook time to 30 minutes.
When cook time is up, allow pressure to naturally release for 5 minutes then release remaining pressure using the quick-release method (i.e. flip the vent valve to ‘Venting’ position).
When all of the pressure is released and the lid unlocks, remove the chicken and shred with two forks or a hand mixer.
On the stove, add olive oil and dairy free milk in a saucepan over medium heat. Whisk continuously until well combined and heated through. Season with salt and pepper.
Add the shredded chicken back to the Instant Pot along the whisked dairy free milk and olive oil mixture. Squeeze in 1/2 lemon juice and stir to combine.
Serve immediately. Garnish with fresh parsley, lemon slices, fresh thyme and shredded or shaved parmesan, if desired.