Preheat oven to 425°F.
In a Dutch oven or large lidded pot, combine chicken breast, corn, carrots, and peas. Add enough water to cover and bring to a boil. Cover the Dutch oven and immediately reduce to a simmer. Simmer for 15 minutes or until the chicken is fully cooked through. Transfer chicken to a cutting board and strain veggies. Set aside in a bowl. Shred chicken with two forks and set aside.
In the same Dutch oven or saucepan, melt butter over medium heat. Add onions and cook until soft and translucent, about 3-4 minutes. Stir in flour, salt, and pepper.
Slowly stir in chicken broth and milk (1/2 cup each per addition, stirring in between). Simmer over medium-low heat until the mixture has thickened, about 7-10 minutes. Remove from heat and set aside.
Place the bottom pie crust in a 9-inch cast iron skillet or pie dish (see note below for skillet). You want enough dough to have the crust come to the top of the sides of the skillet.
Place a layer of shredded chicken in the bottom pie crust. Top with an even layer of vegetables and then pour the hot gravy liquid over the entire pie.
Cover with top crust, trim excess dough, or roll/curl the dough directly under it and crimp edges to seal. This makes for a puffier crust (see note below on crust tips).
Brush the top crust all over with egg/water mixture (this is optional but it's what helps make the crust shiny and golden). Bake for 30 to 35 minutes, or until pastry is golden brown. Let cool for 10 minutes before serving so the filling can stiffen.