Rinse the wild rice blend in a fine mesh strainer under cold running water for about 1 minute until the water runs mostly clear. Dice the onion, slice the carrots and celery ¼ inch thick, and mince the garlic.
Season the chicken breasts with the kosher salt and black pepper and lay them in an even layer in a 6-quart slow cooker. Scatter the rinsed rice, onion, carrots, celery, and garlic over the chicken, add the thyme and bay leaves, then pour in the chicken broth and stir gently so the rice is submerged.
Cover and cook on LOW for 6 hours or on HIGH for 3 to 4 hours without lifting the lid. Taste a few grains of rice at the 6-hour mark; they should be tender with a slight chew, and if still firm, cook another 30 to 60 minutes.
Transfer the chicken to a cutting board, confirm the thickest part reads at least 165°F (74°C), and shred with two forks or meat claws. Discard the bay leaves, return the shredded chicken to the slow cooker, and stir it into the soup.
Melt the butter in a medium saucepan over medium heat, whisk in the flour, and cook for 1 minute until lightly nutty and the texture of wet sand. Slowly whisk in the half-and-half and cook 3 to 4 minutes until thickened, then ladle in about ½ cup (120 ml) of hot broth from the slow cooker and whisk until smooth.
Pour the thickened mixture into the slow cooker and stir until the broth turns creamy and opaque. Cover and cook on HIGH for 15 minutes, then taste and add more salt or pepper as needed, and ladle into bowls topped with chopped parsley.