Heat olive oil in a large pot or Dutch oven over medium heat. Add onion, carrots, and celery. Cook until softened, about 5 to 7 minutes. If using mushrooms, add them now and cook until they release their liquid.
Stir in the wild rice blend, chicken broth, water, thyme, and rosemary. Bring the mixture to a boil, then reduce heat to low, cover, and simmer for 35 to 45 minutes, or until the rice is tender.
While the soup simmers, prepare the roux. In a separate small saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1 minute, stirring constantly. Do not let it brown.
Slowly whisk the half-and-half or milk into the flour and butter mixture until smooth. Cook, stirring, until the mixture thickens slightly, about 3 minutes.
Remove the soup pot from the heat. Stir in the shredded chicken and the thickened cream mixture until fully combined.
Return the pot to low heat. Stir in the salt and pepper. Heat through gently for 5 minutes, being careful not to let the soup boil after adding the dairy.
Taste and adjust seasonings before serving hot.
Notes
For a slow cooker adaptation, combine all ingredients except the flour, butter, and milk/half-and-half in the slow cooker. Cook on low for 6 to 8 hours. Prepare the roux and cream mixture separately, then stir into the slow cooker during the last 30 minutes of cooking.
Use leftover rotisserie chicken for quick preparation.
If you prefer a thicker soup, increase the flour and butter used for the roux by 1 tablespoon each.