Melt the butter in a large pan.
Add the onions, celery, and carrots and cook until tender, about 10-15 minutes.
Mix in the flour and let it cook for a minute.
Mix in the chicken stock followed by the milk.
Reduce the heat and simmer for 10 minutes.
Add the chicken, thyme, green beans/peas, & corn.
Season with salt and pepper.
Pour the filling into a pie dish with the bottom layer.
Place the pie dough over the pie plate and fold any excess under itself. Seal the top and bottom crusts. Cut 3 1 inch slits on the top.
Mix the egg white and the water and brush it on the pie crust.
Place the pie onto a baking sheet and bake in a preheated 400F oven until the crust is golden brown, about 25 minutes.
