Preheat oven to 425 degrees Fahrenheit.
In a large saucepan or skillet melt the butter over medium heat. Add the chopped onions and cook until soft and translucent (about 1-2 minutes).
Remove from heat and add the flour, salt, and pepper; stir well.
Add the chicken broth very slowly, stirring constantly until smooth; mix in the milk.
Return to the stove and cook over medium heat until it boils; stirring constantly. Reduce the heat and simmer for 1 minute longer while continuing to stir (mixture should be thick). Remove from heat.
Mix in the vegetables and chopped chicken until well combined.
Line a 9-inch pie plate with 1 layer of crust, trimming the excess around the edge if necessary. Pour the filling into the crust.
Before adding the top crust, cut slits in the dough for ventilation, then add that layer of crust over the filling. Seal the edges of the crust by crimping with a fork or fold the top crust edges under the bottom crust and press together to seal tightly. To prevent the edges of the crust from burning, cover only the edges with a ring of tinfoil.
Bake for 30-35 minutes in the preheated oven at 425 degrees Fahrenheit or until the top is golden brown. Remove from the oven and let sit for 5 or 10 minutes to cool. Slice and serve.