Melt butter in inner steel pot of pressure cooker set to Sauté on Normal.
Sauté celery, carrots, and onion in the hot butter until tender, 3 to 5 minutes. Add mushrooms and garlic; sauté for an additional 1 minute. Season mixture with salt and black pepper.
Press Cancel to turn off Sauté mode.
Add chicken thighs, wild rice, chicken stock, rosemary sprigs, and bay leaves to vegetable mixture.
Lock pressure cooker lid in place and set steam vent to Sealing. Select Pressure Cook (Manual) and cook for 15 minutes on High pressure.
Allow pressure to release naturally.
Remove rosemary sprigs and bay leaves from pressure cooker and discard.
Transfer chicken thighs to a bowl or plate and use two forks to shred the meat. Return the shredded meat to the pot; stir in parsley and lemon juice. If you’re soup seems watery, you can either turn Sauté mode back on and cook it until thickened to your liking, OR, know that it will thicken up a lot as the rice soaks up more of the liquid.