Poach the Chicken: Place the chicken breasts in a medium saucepan and cover them with cold water by 3cm / 1.2in. Bring the water to a simmer over medium heat, then reduce to medium-low and cook for 18 minutes until the thickest part of each breast reaches an internal temperature of 74°C / 165°F on an instant-read thermometer. Transfer the breasts to a cutting board, let them rest for 5 minutes, then shred or chop them into 2cm / 0.8in pieces. Poaching keeps the chicken moist and produces tender chunks that hold their texture in the finished filling.
Saute the Vegetables: Melt 2 tablespoons of butter in a 10-inch oven-safe skillet over medium heat until the butter foams and the foam subsides, approximately 2 minutes. Add the diced onion, sliced carrots, and sliced celery to the skillet. Cook the vegetables, stirring every 2 minutes, for 8 minutes until the onions turn translucent and the carrots soften enough to pierce with a fork. Add the minced garlic and cook for 1 minute more until fragrant. Softening the vegetables before adding the liquid prevents a raw, crunchy texture in the finished pie.
Build the Gravy: Sprinkle 60g / 0.5 cup of flour evenly over the softened vegetables in the skillet. Stir constantly with a wooden spoon for 2 minutes over medium heat until the flour coats every piece of vegetable and loses its raw smell. Pour in the chicken broth in a slow, steady stream while stirring, then pour in the milk. Continue stirring for 4 minutes until the sauce thickens enough to coat the back of a spoon and no lumps remain. Cooking the flour in the fat before adding liquid creates a roux that thickens the gravy without forming clumps.
Finish the Filling: Add the shredded chicken, thawed peas, kosher salt, black pepper, dried thyme, and garlic powder to the thickened gravy in the skillet. Stir everything together over medium-low heat for 3 minutes until the filling is uniformly combined and heated through. Taste the filling and adjust the salt level before adding the crust. Remove the skillet from the heat and let the filling cool for 5 minutes so the steam does not make the pastry soggy on the bottom.
Preheat and Prepare Pastry: Preheat the oven to 200°C / 400°F with the rack positioned in the center. Unroll the thawed puff pastry sheet on a lightly floured cutting board. If the pastry is smaller than your skillet, use a rolling pin to extend it to a size that overhangs the skillet rim by 2cm / 0.8in on all sides. Cut 4 small slits in the center of the pastry sheet with a paring knife to allow steam to escape during baking.
Top and Seal the Pie: Drape the prepared pastry sheet over the skillet so it covers the filling completely and the overhang folds down over the rim. Press the overhanging pastry firmly against the outside of the skillet rim to seal the crust in place. Brush the entire top surface of the pastry with the beaten egg using a pastry brush. The egg wash creates a deep golden-brown color and a glossy finish during baking.
Bake the Pot Pie: Place the skillet on the center rack of the preheated oven at 200°C / 400°F and bake for 28 minutes until the pastry is deep golden brown across the entire surface and the filling visibly bubbles through the steam vents. Remove the skillet from the oven using oven mitts and let the pot pie rest on a wire rack for 10 minutes before serving. Resting allows the filling to set slightly so it does not pour out when sliced.
