Lightly coat a 5-quart or larger slow cooker with nonstick spray. Heat the olive oil in a large, nonstick pan over medium high. Add the carrots, celery, potato, and onion, and cook until the vegetables are beginning to soften, about 8 to 10 minutes. Stir in the poultry seasoning, salt, and pepper and cook 30 additional seconds. Remove from the heat and transfer to the bottom of the slow cooker.
Rinse the wild rice, then add to the slow cooker with the vegetables. Arrange the chicken breasts on top, then pour in 4 cups of the chicken stock. Cover and cook on low for 6 to 7 hours or high for 2 1/2 to 3 1/2 hours, until the rice is tender and the chicken is cooked through. Remove the chicken to a plate or cutting board and let cool slightly. With two forks or your fingers, shred the chicken, then return to the slow cooker. Place on low to keep warm. If the chicken finishes cooking before the rice, remove it first so that it does not overcook, then re-cover the slow cooker and continue cooking until the rice is tender.
Once the chicken and rice are cooked, melt the butter in a medium saucepan over medium heat. Once the butter is completely melted, sprinkle the flour over the top. Whisk and cook until the flour is golden brown, about 1 minute. Pour in the 2 3/4 cups milk a few splashes at a time, whisking as you go to remove any lumps. Bring to a gentle boil, then reduce the heat to a simmer. Let cook, stirring often, until the mixture thickens, about 4 minutes.
Pour the milk mixture into the slow cooker and stir to combine. If you’d like a thinner soup, stir in additional chicken stock until your desired consistency is reached. Taste add additional salt and/or pepper to taste. Serve hot.
Store leftovers in the refrigerator for up to 5 days or freeze for up to 3 months. Let thaw overnight, then reheat in the microwave or on the stovetop with additional chicken stock, milk, or water to thin the soup as desired.
To make dairy free: Substitute the butter for vegan butter and the milk for unsweetened soy milk or almond milk. Note that the mixture may not be as creamy or may take longer to thicken.
To make gluten free: Omit the flour in step 3. With a fork, firmly whisk 1/4 cup of cornstarch into the milk before adding the milk to the melted butter. Stir to combine and let simmer, stirring often, until thickened.