Preheat the oven to 375 degrees F (190 degrees C). Place the butter in the bottom of a 9x13-inch baking dish, then set the dish in the oven until the butter is fully melted.
Once the butter has melted, arrange the shredded chicken evenly over it in a single layer. Scatter the frozen mixed vegetables evenly across the chicken.
In a mixing bowl, combine the Cheddar bay biscuit mix, the included seasoning packet, and the milk. Stir until the mixture is thoroughly blended.
Pour the biscuit mixture over the chicken and vegetables. Gently spread it across the surface in an even layer, taking care not to stir it into the ingredients beneath.
In a separate bowl, whisk the cream of chicken soup with the chicken stock until smooth.
Pour the soup and stock mixture over the biscuit layer without stirring.
Bake uncovered in the preheated oven until the top is deeply golden brown and the center has only a slight jiggle, about 45 minutes.
Sprinkle the parsley over the cobbler, if using. Allow the dish to stand for 10 minutes before serving.