Preheat the oven to 350°F.
Heat 1/2 tablespoon butter in a 12-inch nonstick skillet over medium-high heat.
Add mushrooms and cook stirring occasionally, until all moisture is evaporated for 5 to 6 minutes.
Place the cooked mushrooms in a bowl and set aside.
Reduce the heat to medium.
Using the same skillet, melt 11/2 tablespoons of butter until foaming. Add flour, whisking constantly, cook for 2 minutes or until bubbling.
Remove from heat.
Slowly first add milk and then chicken stock, whisking constantly, until mixture is smooth, 1 to 2 minutes.
Return to medium heat.
Cook, stirring with a wooden spoon, for 13 to 15 minutes until the sauce thickens and comes to a boil. It should be silky enough to coat the back of a wooden spoon.
Reduce the heat to low.
Add in the shredded chicken and mushrooms in the béchamel sauce and give it a quick stir until all the mushrooms and shreds of chicken are coated with the sauce.
Place the mixture in an ovenproof glass dish (*please see note) and sprinkle cheese evenly on top. If desired, sprinkle some cayenne pepper as well.
Bake it in the oven for 35-40 minutes.
Wait for it to cool, 15 – 20 minutes, prior to serving.