Melt the dairy free butter in a large Dutch oven over medium heat. Add the onion, carrots, and celery and cook for 6 minutes until softened.
Add the mushrooms and cook for 5 minutes until browned. Stir in the garlic, thyme, and rosemary for 30 seconds.
Sprinkle in the flour and stir for 1-2 minutes, then slowly whisk in the chicken broth until smooth.
Add the wild rice blend, whole chicken breasts, bay leaf, salt, and pepper. Bring to a boil, then cover and simmer on low for 20-25 minutes until the chicken reaches 165°F.Transfer the chicken to a plate and shred it with two forks.
Continue simmering the soup, covered, for another 20-25 minutes until the wild rice is tender and starting to split open.
Stir in the oat milk and shredded chicken and heat gently for 5 minutes. Discard the bay leaf, stir in the lemon juice, and adjust the salt to taste.