Meanwhile, melt 2 tablespoons butter in a large Dutch oven or braising pot over medium/medium-high heat. Salt chicken on both sides with 1/2 teaspoon sea salt, then sear in butter until golden with an internal temperature of 165 degrees F, 5 - 7 minutes per side. Set cooked chicken breasts on a rimmed plate and set aside. When cool enough to handle, use two forks to gently shred chicken.
In the same Dutch oven, add wine and use a wooden spoon to scrape up any brown bits. Add remaining 2 tablespoons butter, and when melted, add the onion, celery, and carrots. Sprinkle with a 1/2 teaspoon sea salt and the pepper, and cook, stirring often, until tender, about 6 minutes. Add the garlic and cook one minute more. Fold in flour and allow it to brown slightly. Drizzle in broth, adding a total of 5 cups. Fold in kale ribbons and add 2 sprigs thyme (stem and all, if using). Gently simmer soup over medium-low heat and cook 20 minutes. Be sure to stir frequently – soup may stick to the bottom as it cooks. As soup cooks and thickens, add up to 1 more cup broth as needed.
Fold in cream, cooked wild rice blend, and chicken. Add sea salt and pepper to taste. Turn heat to the lowest setting and stir gently until heated through. Ladle into bowls, garnish with parsley and thyme leaves, a few turns of black pepper, and serve.