Cook mushrooms: Melt ½ tbsp butter in a pan. Add mushrooms with a pinch of salt. Cook 6–8 minutes until liquid evaporates and mushrooms are lightly golden. Set aside.
Make the base: Melt 2 tbsp butter in a pot. Add 2 tbsp flour and whisk 1–2 minutes.
Slowly add 4 cups broth, whisking. Bring to a gentle simmer.
Add fillings: Add shredded chicken and mushrooms. Simmer 5 minutes.
Finish: Lower heat and add 1 cup heavy cream. Heat 2–3 minutes (do not boil). Season with salt and black pepper to taste.