Food & Wine offers a recipe for chicken and wild rice soup that calls for celery, carrots, onion, garlic and thyme. The recipe uses a combination of water and chicken stock as a base and heavy cream and all-purpose flour as a thickener.
To prepare chicken and wild rice soup, melt 4 tablespoons of unsalted butter in a large saucepan. Cut 3 celery ribs and 2 carrots into 1/2-inch pieces, and add the cut vegetables to the pan along with 1 chopped medium onion and 2 minced garlic cloves. Sprinkle in 1 1/2 teaspoons of finely chopped thyme and a generous pinch of salt and pepper, and then cook over medium heat while stirring occasionally. Once the vegetables begin to soften, sprinkle 1/4 cup of all-purpose flour over them, and stir for about three minutes until lightly browned.
Add 1 cup of wild rice to the saucepan along with 2 quarts of chicken stock and 2 cups of water. Bring the mixture to a boil, reduce the heat, and simmer for about 30 minutes. Add 4 cups of roasted chicken or turkey meat chopped into bite-size pieces, and simmer for 10 to 15 more minutes until the rice is tender. Mix in 1 cup of heavy cream, and season to taste with salt and pepper before serving.