(Note: Be sure to cook your wild rice blend and chop all your veggies ahead of time.)
Saute Veggies: In a large Dutch oven or pot, melt 1 tablespoon of the butter over medium-high heat. Add the onion, celery, carrots, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook, stirring frequently, just until the vegetables are tender, 4 to 5 minutes. Add the garlic for the last 30 to 60 seconds.
Start the Soup: Stir in the broth, poultry seasoning, remaining 1 teaspoon salt, remaining 1/2 teaspoon pepper, and the chicken breasts.
Braise the Chicken: Bring to a boil and then reduce heat to a simmer. Simmer for 12-15 minutes, until chicken is fully cooked. (Stir the chicken breasts around occasionally for even cooking.) Remove and set aside the chicken to a cutting board to rest.
Make the White Sauce: While the chicken cooks, make the white sauce that will thicken the soup. In a medium pot, melt the remaining 7 tablespoons of butter over medium heat. Stir in the flour and cook for 1-2 minutes, or until bubbly. Slowly pour in the milk while continually whisking. Increase the heat to medium-high and whisk constantly until the milk mixture thickens to the consistency of thin gravy. Usually takes 3-5 minutes.
Add It All Together: Add the white sauce* and prepared wild rice to the soup pot, stir, reduce the heat to medium low. Using two forks, shred the chicken. Return the chicken to the pot and heat until warmed through, about 5 minutes. Taste and adjust the seasoning if needed.
*Note: The white sauce will continue to get thicker the longer it sits, so it's important to add it right away. Thin out the soup with more broth, if needed.