Chop and prepare all the vegetables.
Put the instant pot on the saute setting and lightly drizzle the bottom of the pot with some olive oil. Saute the onion, celery, and carrot for about 5 minutes until the onions become tender.
Add the garlic and mushrooms. Give it a stir to prevent it all from sticking.
Top the veggie mixture with the wild rice, chicken breasts, thyme, bay leaf, oregano, salt, and pepper. Gently stir again. Then, pour in the chicken broth and water.
Cover the instant pot and cook it on high pressure in the “sealing” setting for 35 minutes.
After the time is up, allow it to sit for 5 to 10 minutes, then do a quick release by carefully moving the lever to the “venting” position with a kitchen towel to block the steam.
Once the pressure is fully released, remove the lid. Carefully pull out the bay leaf and chicken breasts. Throw away the bay leaf and place the chicken in a separate dish. Shred the chicken with two forks.
In a small bowl, whisk the eggs. Slowly whisk in a small ladle of hot soup broth to temper the eggs. This process warms them without cooking or scrambling them. Once the broth is incorporated into the eggs, add the mixture into the rest of the soup. Mix it in to make the broth smooth and creamy.
Top the soup off with the frozen peas (they will defrost in the soup), as well as the lemon juice and shredded chicken. Then, adjust the salt and pepper to your taste, serve in individual bowls, and enjoy!