Rinse the wild rice under cold water and drain well. Dice the onion, slice the carrots, celery, and mushrooms, and mince the garlic. Pat the chicken dry and season both sides with ½ teaspoon salt and ¼ teaspoon pepper.
Press Sauté on High and melt the butter. Cook the onion, carrots, and celery for 4 minutes, add the mushrooms and cook 3 minutes, then stir in the garlic for 30 seconds. Sprinkle in the flour and stir for 1 to 2 minutes until no dry flour remains.
Pour in 1 cup of the chicken broth and scrape the bottom of the pot firmly with a wooden spoon to loosen every browned bit and flour paste. Press Cancel.
Add the remaining 3 cups broth, the water, rinsed wild rice, thyme, sage, bay leaves, remaining ½ teaspoon salt, and remaining ¼ teaspoon pepper. Stir once, then lay the seasoned chicken breasts on top in a single layer without stirring them in.
Lock the lid, set the valve to Sealing, and cook on High pressure for 30 minutes. Let the pressure release naturally for 10 minutes, then vent the remaining pressure.
Remove and discard the bay leaves. Transfer the chicken to a cutting board and shred it with two forks or shredder claws, then return it to the pot. The rice should be split open and tender with a chewy center.
Press Sauté on Normal and stir in the heavy cream. Simmer gently for about 5 minutes until slightly thickened, avoiding a hard boil. Press Cancel, then stir in the lemon juice and 1 tablespoon of the parsley. Taste and add salt as needed.
Let the soup rest for 5 minutes, then ladle about 1½ cups into each bowl. Garnish with the remaining 1 tablespoon parsley and serve hot.