Heat 1 tablespoon butter in large Dutch oven over medium-high heat. Add mushrooms and sauté until browned and liquid is released. Remove to plate and set aside.
To same pot, melt 3 tablespoons butter and add carrots, onion and celery, sprinkle with a little salt and cook until onion is translucent and vegetables start to soften. Add garlic, thyme and bay leaf, sauté another minute until fragrant.
Sprinkle flour over vegetables, cook and stir constantly for 2 minutes.
Stir in wild rice, then gradually whisk in chicken stock. Stir in the reserved mushrooms and bring to a boil, then turn heat down and simmer for about 45 minutes, stirring frequently, until rice is tender.
Add the cooked and cubed chicken and stir to combine and heat through, about 5 minutes.
Whisk in the heavy cream, simmer gently to heat through, about 3 to 5 minutes.
Season to taste with salt and pepper and serve immediately.