Using a pastry blender or two forks, cut in butter until mixture forms small pea-sized balls.
Slowly add water and toss mixture lightly with a fork until dough forms a single ball. If needed, add more water in very small increments until no dry ingredients left in bowl.
Divide dough into 2 equal sized pieces and chill for at least 20 minutes. (Dough can be wrapped and chilled overnight if needed.)
Flour table well and roll out first piece of dough until approximately 1/8 inch thick. Use caution at this step to ensure that dough does not stick to rolling pin or table. (Dust pin and table if dough begins to stick.) The dough should be approximately 12" in diameter.
Roll finished dough onto floured pin and unroll into pie dish.
Fill pie crust (see instructions below for making the filling).
Roll second piece of dough and place over filled pie. Crimp edges of the two pieces together.
Brush top of entire pie crust lightly with beaten egg. This will give your crust a nice golden color.
Bake at 375 degrees for approximately 40-45 minutes, or until crust is golden in color.
For the Filling
Place the carrots, peas, celery, and corn in boiling water for approximately 10 minutes. Remove from heat and drain. Add the chicken and mix well.
Meanwhile, melt the butter and saute onion until translucent (about 6-8 minutes).
Add flour, salt, pepper and other seasonings (if desired). Add milk and chicken broth and stir until mixture is well combined and has thickened slightly.
Combine chicken and vegetable mixture with chicken broth mixture and stir.