Rinse the uncooked wild rice in a fine-mesh strainer under cold running water for about 30 seconds, then set aside to drain. If using pure wild rice, consider soaking it in cold water for 30 minutes to speed up cooking.
Dice the onion, carrots, and celery into small, even pieces. Mince the garlic. Having all your vegetables prepped before cooking ensures a smooth process.
Melt the butter in a large Dutch oven over medium-low heat. Add the diced onion, carrots, and celery, and cook low and slow for 8 to 10 minutes until softened. Stir in the minced garlic and cook 1 to 2 minutes until fragrant.
Add the dried thyme, rosemary, and poultry seasoning to the pot, stirring for about 30 seconds to bloom the herbs. Season lightly with salt and pepper.
Sprinkle the flour over the vegetable mixture and stir constantly for 1 to 2 minutes to form a blonde roux, cooking out the raw flour taste.
Gradually pour in the chicken broth, whisking to prevent lumps. Add the bay leaf and rinsed wild rice, stir to combine, and bring to a gentle boil.
Nestle the chicken breasts into the broth, reduce heat to medium-low, and cover. Simmer for 35 to 45 minutes until the wild rice is tender and the chicken reaches 165°F (74°C).
Remove the chicken from the pot, discard the bay leaf, and shred the chicken into bite-sized pieces using two forks.
Reduce the heat to low and slowly stir in the heavy cream until fully incorporated and the broth is rich and creamy.
Return the shredded chicken to the pot and simmer for 5 more minutes to warm through. Taste and adjust seasoning with additional salt and pepper. Garnish with fresh parsley and serve.