Melt the butter in a pan over medium-high heat, add the mushrooms and cook until the mushrooms have released their liquids and the liquid has evaporated, about 10-15 minutes, before setting aside.
Melt the butter in the pan, add the onions, carrots and celery and cook until tender, about 8-10 minutes.
Mix in the garlic and thyme and cook until fragrant, about a minute.
Add the broth, rice, chicken and mushrooms, bring to a boil, reduce the heat and simmer, covered, until the rice is tender, about 20-30 minutes.
Mix in the milk and cheese and cook until the cheese has melted, before seasoning with salt and pepper to taste.
I love the idea of this and like to keep it. My husband has had a big Colon operation and has to eat things like this.