Bring 4 cups salted water to a boil. Add the wild rice and simmer until tender, 45 to 60 minutes. Drain well and set aside; you need about 2 cups cooked rice.
Cover the dried mushrooms with hot water and soak for 15 minutes. Drain, rinse away any grit and chop into chunks. Discard the soaking liquid.
Melt 1 tablespoon butter in a Dutch oven over medium-high heat. Brown the button mushrooms in batches until deeply golden and their moisture has evaporated, 8 to 12 minutes total. Transfer to a plate.
Reduce heat to medium. Add the remaining 3 tablespoons butter, leek, onion, carrots and celery. Cook 6 to 8 minutes, until softened.
Add the garlic, thyme and chopped dried mushrooms. Stir for 1 minute, until fragrant.
Add the wine and scrape up the browned bits. Simmer until the wine has almost completely evaporated.
Stir in the flour and cook 1 to 2 minutes, stirring constantly.
Gradually add the turkey broth, whisking until smooth. Return the browned mushrooms to the pot.
Simmer gently 20 to 25 minutes, stirring occasionally, until the vegetables are tender and the broth has thickened slightly.
Stir in the cooked wild rice, turkey and milk. Heat gently 5 to 10 minutes until warmed through. Do not let the soup boil.
Stir in the cream and season with salt and pepper to taste.
Garnish with parsley and serve hot.