Heat 2 tablespoons of butter in a 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.

Add chicken stock, shredded chicken, wild rice, mushrooms, carrots, celery, potatoes, and Old Bay seasoning. Stir to combine.

Reduce heat and simmer uncovered for 45 minutes, stirring occasionally, until the rice is tender.

In a separate saucepan, melt remaining butter over medium-high heat. Whisk in flour and cook for 1 minute to make a roux.

Gradually whisk in the milk, stirring until the roux thickens.

Add the roux mixture to the soup, season with salt and pepper, and stir well. Serve warm.