Melt butter in a large stockpot or Dutch oven over medium heat. Add mushrooms, onion and celery. Cook, stirring occasionally, until tender, about 8-10 minutes.
Stir in garlic and thyme until fragrant, about 1 minute.
Stir in chicken, chicken stock, wild rice, bay leaf, 1 teaspoon salt and 1/2 teaspoon pepper. Bring to a boil; reduce heat and simmer, covered, until flavors have blended, about 20 minutes.
Stir in butternut squash or sweet potato; simmer, covered, until the chicken is cooked through and the rice is tender, about 15-20 minutes.
Remove chicken from the Dutch oven; shred, using two forks.
Stir in chicken and kale until wilted, about 2 minutes. Stir in parsley; season with salt and pepper, to taste.