Heat the olive oil in a large pot over medium heat. Add the onion, carrots, and celery and cook for 5–6 minutes, until they begin to soften.
Add the mushrooms and cook for another 5 minutes, stirring occasionally, until softened and lightly browned.
Add the garlic, thyme, rosemary, salt, and pepper. Cook for another 30 seconds, stirring frequently.
Add the wild rice and chicken broth. Bring to a boil, then reduce the heat, cover, and simmer for 30–40 minutes, or until the rice is tender. Cooking time may vary depending on the wild rice blend used.
Add the shredded chicken and milk and stir to combine. Simmer gently for another 3–5 minutes.
In a small bowl, whisk together the cornstarch and water. Stir into the soup and simmer for 2–3 minutes, until slightly thickened.
Reduce the heat to low. Place the Greek yogurt in a bowl and stir in a few spoonfuls of the warm broth to temper it. Gradually stir the yogurt mixture back into the soup. Do not allow the soup to boil once the yogurt has been added.
Stir in the spinach and lemon juice. Cook for another 1–2 minutes, until the spinach has wilted. Taste and adjust the seasoning as needed.
Divide between bowls and garnish with fresh parsley, if desired.