Sauté the mushrooms in 1 T butter until browned (6-8 minutes).
While the mushrooms are sautéing, melt 1/4 c butter in a medium sauce pan over medium heat. Once the butter is melted and bubbly, add the flour and whisk until combined (about 1 min).
Add milk and stir until thickened (about 4 min). Once thick, add the chicken broth and continue to stir until the sauce is desired texture (about 3 min). Add salt and pepper.
Add the mushrooms and chicken to the sauce and stir. Cook the microwavable rice. Serve immediately over rice.