Cook the wild rice. Prepare the wild rice according to the package directions, which usually takes 45 to 60 minutes, until the grains split and stay chewy; drain. This can be done 1 to 2 days ahead and kept chilled until you are ready to make the soup.
Sauté the vegetables. Melt the butter in a large Dutch oven or heavy pot over medium heat. Add the onion, celery, carrots, mushrooms, and garlic, and cook, stirring, for about 8 minutes, until the onion has softened.
Make the roux. Sprinkle the flour over the vegetables and stir it in, cooking for 1 minute. Slowly add the broth, stirring constantly, then raise the heat to medium-high until the soup comes to a low boil.
Simmer. Reduce the heat to medium and add the cooked wild rice, parsley, salt, pepper, thyme, rosemary, and shredded turkey. Simmer for 20 minutes.
Finish. Stir in the milk, half-and-half, and lemon juice, and heat for 3 to 5 minutes, keeping the soup just below a boil — never let it boil once the dairy is in. Taste and adjust the salt and pepper as needed. Serve immediately.
