Rinse the wild rice under cold running water for about 30 seconds. Add the rinsed rice, 3 cups of water, and 1/2 teaspoon of salt to a medium saucepan. Bring to a boil over high heat, then reduce heat to low, cover, and simmer for 45 to 55 minutes until the grains split open and the rice is tender with a slight chew. Drain any extra water, fluff with a fork, and set aside (you should have about 2 1/2 cups of cooked wild rice).
Dice the onion, carrots, and celery into small, even 1/4-inch pieces. Mince the garlic and set aside. Chop or shred the cooked turkey into bite-sized pieces and measure out 3 cups. Measure the half-and-half and let it sit on the counter to lose its chill. Chop the parsley and set aside for the end.
Set the Dutch oven over medium heat and add the 6 tablespoons of butter. When the butter has melted and stopped foaming, add the onion, carrots, and celery. Sprinkle in 1/2 teaspoon of the kosher salt to help the vegetables release their moisture. Cook for 6 to 8 minutes, stirring often, until the onion is soft and translucent. Add the garlic and stir for 30 to 60 seconds, just until fragrant.
Sprinkle the 1/3 cup of flour evenly over the vegetables. Stir constantly for about 2 minutes, until the flour disappears into the butter and smells lightly toasty. Keep the heat at medium so the mixture stays pale and does not brown.
Pour in the broth about 1 cup at a time, stirring or whisking well after each addition to prevent lumps. Add the bay leaf, dried thyme, and black pepper. Raise the heat and bring the soup to a gentle boil, stirring often. Reduce the heat to medium-low and simmer for 10 to 15 minutes, until the soup has thickened slightly and the carrots are tender.
Stir the cooked wild rice into the pot. Add the 3 cups of cooked turkey and stir to combine. Simmer gently for about 10 minutes so the flavors can come together and the turkey warms through. If the soup looks too thick, stir in a splash of broth or water.
Turn the heat down to low. Slowly pour in the 1 1/2 cups of half-and-half while stirring. Warm the soup for 5 minutes without letting it boil, which keeps the cream smooth. Fish out the bay leaf and discard it. Taste, then stir in the remaining 1/2 teaspoon of kosher salt and more if needed.
Ladle the hot soup into bowls. Sprinkle each bowl with fresh parsley. Serve right away while it is steamy and creamy.