Melt the butter during a pan over medium-high heat, add the mushrooms and onion and cook till the mushrooms have discharged their liquids and therefore the liquid has gaseous, regarding 10-15 minutes, before setting aside.
Melt the butter within the pan, add the onions, carrots and celery and cook till tender, regarding 8-10 minutes.
Mix within the garlic and thyme and cook till aromatic, a few minute.
Add the broth, rice, chicken and mushrooms, bring round a boil, cut back the warmth and simmer till the rice is tender, regarding 20-30 minutes.
Mix within the milk and cheese and cook till the cheese has thawed, before seasoning with salt and pepper to style.