Mince the onion and garlic. If you’re using vegetables like carrot and celery, dice them finely so they soften quickly and blend nicely into the broth.
In a large pot, melt the butter over medium heat. Add the onion and carrot (and celery, if using) and sauté until the vegetables are softened and the onion is translucent, about 4–5 minutes.
Add the garlic and mushrooms (if using) and cook for another 1–2 minutes, until the mushrooms release a bit of moisture and become fragrant.
Stir in the flour (if using) to lightly coat the vegetables and cook for 1 minute to remove the raw flour taste.
Pour in the chicken broth and milk, whisking to prevent lumps if you used flour. Bring the mixture to a gentle simmer.
Add the cooked wild rice, shredded chicken, and thyme. Simmer gently for 8–10 minutes, letting flavors meld and the rice heat through.
If you’re using cream cheese, whisk it in now until fully incorporated and smooth. You can substitute cream or half-and-half for a lighter version.
Season with salt and pepper to taste. If the soup is too thick, thin with a bit more broth or milk until it reaches your preferred consistency.
For a silkier finish, you can blend 1/4 of the soup and stir it back in, or simply use an immersion blender to puree a portion and leave some texture.
Taste once more and adjust seasoning. Serve hot with crusty bread or a fresh green salad for a complete meal.