Preheat oven to 425 degrees F
Bring 4 cups of water to boil. Sprinkle with a little salt and once boiling, add the carrots and celery. Blanch for 2-3 minutes, drain and set aside.
In a large skillet, melt the butter over medium-high heat. Once melted, lower the temperature to medium heat. Add the onion and saute until tender and translucent. Add the garlic, sauteeing for 30 seconds, or until fragrant. Sprinkle with the flour, salt, and pepper, and mix to combine.
Slowly pour in the chicken stock first, whisking consistently, followed by the milk (or cream). Stir constantly and allow the sauce to thicken, about 4-5 minutes, reducing the heat if needed to prevent burning. Once thickened, add the cooked chicken, frozen peas and corn, and the blanched carrots and celery. Mix to combine then remove from heat. Allow the creamy chicken mixture to cool slightly, about 5-10 minutes.
Add the chicken mixture to the pie pan already lined with the bottom crust. Top with the second circle of dough. Press the overhang of the two crusts together and fold the edge of the crusts in half, creating a thicker crust edge. Flute/crimp the edges together and cut slits on the top crust which will allow steam to escape as your pie cooks. Optional - brush the top of the crust with the beaten egg.
Bake for 35-40 minutes, or until crust is golden brown and the internal temperature of the pie is 165 ℉. Serve warm, and enjoy!