How to Make It
Preheat the oven to 375°F. Shred the chicken, reserving the bones.
In a medium heavy pot, bring the chicken bones and milk to a simmer, then simmer 30 minutes. Discard the bones, then stir the broccoli and cream cheese into the milk and simmer, whisking, until the cream cheese is melted, about 3 minutes. Stir in the shredded chicken and transfer to a 3-quart baking dish. Scatter the cheddar cheese over the casserole and bake until the filling is bubbling and the cheddar cheese is melted, about 20 minutes.