Heat oil in a large pot or dutch oven over medium heat. When hot, add the onion, carrot, celery, garlic, salt and pepper. Cook, stirring, until the veggies are just tender, about 5 minutes.
Add the broth, bay leaves, thyme, rosemary, oregano and cayenne. Stir to combine and increase heat so that it comes to a boil.
Once the soup is boiling, reduce the heat to a low simmer over low heat. Add the diced chicken (or turkey) and the uncooked wild rice. Cover and cook until the rice is tender, about 45 minutes. *see note below
After the rice is tender, uncover the soup and increase the heat so that is it simmering more vigorously, but not boiling (about medium heat).
In a small bowl, combine the dairy-free creamer and arrowroot flour and whisk until arrowroot has dissolved in the creamer.
While slowly stirring the soup, slowly pour in the creamer/arrowroot mixture until it is well combined in the soup. Continue to cook the soup, uncovered and simmering, stirring often until it has thickened, 5 to 10 more minutes.
Stir in the lemon juice, serve and enjoy!