Add 1¼ cups chicken stock, 1 teaspoon olive oil, and ½ teaspoon salt to a small saucepan. Bring to a boil. Add ¾ cup wild rice blend, reduce to a simmer, cover, and cook for about 15 minutes or until the liquid is absorbed. Remove from heat and keep covered.
In a large pot over medium heat, melt 2 tablespoons butter. Add ½ chopped yellow onion and 1 chopped yellow squash. Sauté for 4–5 minutes until soft and lightly golden. Stir in 4 cloves minced garlic during the last minute.Transfer to a blender with 4 cups chicken stock and blend until smooth and velvety. In the same pot, heat 2 tablespoons olive oil over medium-high heat. Add 2 seasoned chicken breasts and cook for 3–4 minutes until golden on the bottom. Flip, then add 4 ounces mushrooms around the chicken. Cook another 3–4 minutes until browned and fragrant.
Remove the chicken and deglaze the pot with a splash of the blended stock, scraping up the browned bits. Stir in 1 tablespoon parsley, 2 teaspoons dill, 1 teaspoon oregano, and 1 teaspoon salt.
Add 3 chopped carrots and 2 chopped celery stalks. Return the chicken to the pot, pour in the remaining blended stock and 2 more cups chicken stock. Cover and simmer for 15 minutes.
Remove the chicken, shred it with two forks, and return it to the pot. Add 2 tablespoons lemon zest and simmer 5 more minutes.
Remove from heat. Once the soup is no longer simmering, slowly stir in 1 cup Greek yogurt until smooth. This is the method that’s worked best for me to keep the soup creamy every time.