Shred the chicken.
Dice onions, carrots and celery. Mince the onions.
Peel, dice and parboil potatoes. Drain and set aside.
Melt butter in a dutch oven or deep sided skillet over medium heat.
Cook onions, carrots, celery and garlic until softened, about 10 minutes.
Whisk in flour and cook, stirring constantly, for about 5 minutes.
Add chicken broth and milk. Whisk, cooking until thickened.
Remove from heat.
Preheat oven to 375°.
Stir in shredded chicken and potatoes.
Rinse corn and peas and stir into mixture.
Spoon filling into 2 deep dish pie dishes.
Cut puff pastry into strips and make a lattice pattern by overlapping the strips.
Beat an egg with 1 tablespoon of water.
Brush over puff pastry.
Place pot pie on a rimmed baking sheet and bake for 45 minutes to 1 hour until puff pastry is golden brown and filling is bubbly.
Remove from oven and allow to rest about 10 minutes before serving.