Set your oven to preheat at 425°F before you start prepping the filling.
Bring a small pot of water to a boil, toss in the diced potatoes and sliced carrots, and cook them for 5-8 minutes until just tender. Drain well and set aside.
In a large skillet, melt the butter over medium heat, then toss in the diced onion, letting it soften for about 2-3 minutes.
Season the onions with salt, pepper, thyme, and poultry seasoning, then dust the flour over everything. Stir constantly for a minute so the flour cooks through without clumping.
Slowly pour in the chicken broth while whisking, followed by the milk. Once combined, fold in the drained potatoes and carrots, and let the mixture bubble gently until it thickens into a gravy-like consistency. Taste and adjust seasoning as needed.
Turn off the heat, then mix in the shredded chicken and peas until everything is evenly coated in the gravy.
Lay one pie crust into a deep pie dish, pressing it gently against the bottom and sides. Ladle the warm filling directly into the crust.
Cover with the second crust, trimming away any overhang. Pinch the top and bottom edges together with your fingers to seal, creating a crimped border.
Brush the top generously with the egg wash, then cut four small slits across the surface to vent steam during baking.
Set the pie on a baking sheet and slide it into the oven. Bake for 30 minutes, or until the crust turns a deep golden brown. Letting the baked pie rest for about 10 minutes before slicing helps the filling set so it doesn't run when served.