Melt butter in a dutch oven over medium heat.
Add celery and onions and cook until translucent.
Add garlic and cook an additional 30 seconds
Sprinkle flour over the vegetables and stir to combine. Continue stirring and allow the flour to toast for a minute or two, until it smells nutty.
Pour in chicken broth, milk, chicken, carrots, thyme, salt, and pepper. Allow to cook, stirring occasionally until the sauce has thickened (about 10 minutes).
Turn off heat and allow mixture to cool while you prepare the pie dough.
Preheat oven to 400℉.
Roll out the dough and place one inside the pie pan.
Fill with the pot pie filling and top with the other dough.
Crimp and close edges and with a sharp knife slice the top to vent.
Mix egg and milk and brush over the top of the pie.
Bake for 35 minutes. Allow to cool for 15-20 minutes.