Heat the olive oil in a Dutch oven or large soup pot over medium heat. Add the carrots, celery, and shallots or onions. Season with salt and pepper then saute until the vegetables are tender, 8-10 minutes. Add the garlic, poultry seasoning, and dried thyme then saute until very fragrant, 1-2 minutes.

Add the chicken stock or broth to the pot then bring to a boil. Add the chicken breast halves, cover, and simmer on low until cooked through, 10-12 minutes. Transfer chicken to a cutting board, chop or shred when cool enough to handle, then refrigerate.

Meanwhile, add the wild rice to the pot and stir to combine. Cover and simmer until the rice is al dente, 20-30 minutes, stirring occasionally.

When the rice has about 10 minutes left, melt the butter in a small saucepan over medium heat. Sprinkle in the gluten-free flour and whisk for 1 minute. Slowly drizzle in the milk while whisking to create a smooth sauce, then cook while stirring until thick and creamy, 1-2 minutes. Remove from heat and set aside.

Once the rice is al dente, slowly stream in the butter/flour/milk mixture while constantly whisking the soup. Add the chopped chicken then cook for 5 more minutes, stirring frequently, or until the soup has thickened and the rice is tender. Taste and adjust seasoning if necessary. Remove from heat and let sit for 5-10 minutes to thicken slightly before serving.