To make the crust: In a food processor, add the flour, salt and butter. Mix until the ingredients are combined. With the motor running, slowly add the water-vinegar mixture and mix until the dough begins to gather on itself.
Dump out the dough and form 2 balls. Wrap in plastic and refrigerate for 30 minutes.
Preheat the oven to 350 degrees F.
In a large saucepan, heat the butter until foamy. Add the flour, and chicken base, whisking constantly until golden brown, 5 to 8 minutes. Remove from the heat and whisk in the stock, cream and milk.
Put the pan back on low heat until the mixture has thickened (if the sauce is too thick, add more stock.) Taste for saltiness. Whisk in sage, nutmeg, salt (if necessary), and pepper. Remove from the heat, and in a large bowl, combine with the vegetables and chicken. Set aside.
Spray a 9 by-13 cast iron dish with nonstick cooking spray.
Roll out the first ball of dough slightly larger than the pan. Lay the bottom crust in the pan, letting the edges hang over. Fill with the chicken and sauce mixture.
Roll out the second crust slightly larger than the pan and place on top of the pie. Wet the edges with some water and crimp the edges of the 2 crusts together. Roll the extra over and tuck into the pan.
Mix the egg yolks, cream and water to form a glaze and brush on the top of the pie. Cut a slit in the top to vent the steam and refrigerate for 15 minutes to set.
After setting, place in the oven and immediately turn the temperature down to 250 degrees F. After 45 minutes, increase the heat to 350 degrees F and bake for 15 more minutes or until bubbly.
Let pie rest for 15 minutes before serving.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.