Preheat your oven to 425°F (220°C).
In a large skillet, melt the butter over medium heat. Add the diced chicken to the skillet and cook for about 5-7 minutes, stirring occasionally, until the chicken is no longer pink on the outside.
Add the chopped onion, diced carrots, diced celery, and minced garlic to the skillet with the chicken. Cook for another 5-7 minutes, or until the vegetables are softened and the chicken is cooked through.
Sprinkle the flour over the chicken and vegetables, stirring well to coat. Cook for 1-2 minutes, stirring constantly, to eliminate the raw flour taste.
Gradually stir in the chicken broth and milk, whisking constantly to prevent lumps. Bring the mixture to a simmer and cook for about 5 minutes, or until the sauce thickens.
Stir in the dried thyme, dried rosemary, salt, and pepper. Add the frozen peas to the skillet, stirring until everything is well combined. Remove from heat. Let the filling cool for about 10 minutes.
Place one of the pie crusts into a 9-inch pie dish, pressing it gently into the bottom and up the sides. Pour the chicken and vegetable filling into the prepared pie crust, spreading it out evenly.
Place the second pie crust over the filling. Trim any excess crust, then crimp the edges to seal. Cut a few small slits in the top crust to allow steam to escape.
Brush the top crust with the beaten egg to give it a golden, shiny finish.
Place the pie dish on a baking sheet to catch any drips and bake in the preheated oven for 30-35 minutes, or until the crust is golden brown and the filling is bubbling.
Let the chicken pot pie cool for a few minutes before slicing and serving.