Heat 2 tablespoons butter in large dutch oven over medium-high heat. Add onion and sauté for 5 minutes, stirring occasionally, until translucent and brown.
Stir in the chopped garlic and cook for an additional 1-2 minutes, stirring occasionally, until fragrant.
Add in the chicken stock, chicken, wild rice, mushrooms, carrots, celery, potato, and Old Bay seasoning. Stir to combine.
After the soup reaches a simmer, reduce heat to medium-low then cover and simmer for 45 minutes, or until the rice is tender. Stir occasionally and check on the rice.
In a separate saucepan, cook the remaining butter over medium-high heat until melted. Whisk in the flour until combined, and cook for 1 minute. (this is the roux)
Gradually add in the milk, and whisk until combined. Stir frequently until the roux has thickened.
Add the roux to the soup and stir gently until combined. Season with salt and pepper.