Pre-heat the oven to 350 degrees. Simmer the chicken, veggies and seasoning in a stock. You can boil potatoes while the chicken is cooking. Take the chicken and potatoes out after half an hour, set the potatoes aside, pull the meat off the chicken and return the bones to the stock pot for another hour.
Combine the crust ingredients in a large bowl, working the butter and milk in until a dough ball forms. Let the dough ball rest for at least half an hour. After resting, roll the dough out into rounds that fit into the pie pans. Reserve enough rolled out dough to cover the pies. Put the rounds in pans and add pie weights, then bake for 15 minutes.
Sauté the vegetables just enough to soften them.
Make a roux with the filling flour and buerre vert. Add in broth from the chicken cooking, stirring well to get a smooth thick sauce. Add in the vegetables, potatoes and chicken to the sauce.
Ladle filling into the baked pie crusts, cover with extra dough and brush the dough with egg wash. For an extra flourish, cut shapes from the dough and set the shapes on top of the pie.
Bake the pies for half an hour until the crust turns golden brown. Cool for 10 minutes. Enjoy.