Preheat the oven to 425°F. Make the creamy chicken filling: Melt the butter in a large oven-proof skillet over medium-high heat. Add the onion and celery and cook until tender, about 5 minutes, seasoning with the salt and pepper as the vegetables cook.
Sprinkle in the flour, and stir until thickened. Add the wine and cook until flour begins to dissolve. Add the chicken broth and milk and bring to a boil.
Add the frozen vegetables and shredded chicken and simmer until the sauce is thickened (like gravy consistency). Remove the pan from the heat.
Make the cobbler topping: Stir together the biscuit mix, cheese, and cold water in a medium bowl just until a dough forms. Use a cookie scoop to distribute the biscuit dough in an even layer over the top of the chicken filling. Bake for 20 to 25 minutes or until the biscuits are golden brown and cooked through.
Stir together the melted butter and seasoning mix in a small bowl. Brush the butter evenly over the biscuit layer and sprinkle with parsley. Serve hot.