Preheat oven to 400 degrees F. Line the bottom and sides of a 9- or 10-inch pie plate with 1 pie crust. Fill the bottom with the shredded chicken, peas and corn; set aside.

Place the potatoes and carrots in a large sauté pan or skillet. Fill with cold water, until the vegetables are just covered. Heat over high heat until it comes to a boil. Reduce heat, cover and let simmer for 8 to 10 minutes, until just tender. Drain, then transfer to the pie plate over the chicken, peas and corn.

Wipe out the pan and return to heat. Melt 2 Tablespoons of the butter in the same pan, then add the onions. Cook until soft, about 5-7 minutes. Add the other 3 Tablespoons of butter and toss until melted. Stir in the flour, celery seed, salt and pepper. Slowly mix in the chicken broth and milk. Simmer over medium heat, stirring frequently, until thickened. Season with more salt, to taste. Pour into the pie plate over the chicken and veggies. Do not stir.

Place the other pie crust over top of the filling, then tuck the edges under the bottom crust and pinch to seal. Brush lightly with egg wash. Slice an "X" in the center of the crust to allow steam during baking. Bake for 45 minutes. Remove from the oven and let set for 10 minutes before serving.