Preheat and prep: Heat your oven to 400°F (200°C).
Lightly grease a 9x13-inch baking dish or a deep 9-inch pie dish.
Make the creamy base: In the dish, whisk together the cream of chicken soup, cream of mushroom soup, chicken broth, milk, garlic powder, onion powder, thyme, pepper, salt, and melted butter.
Add the fillings: Stir in the chicken and frozen mixed vegetables until everything is well coated. Smooth the top.
Add the crust: Lay the pie crust (or puff pastry) over the filling. Tuck or trim the edges to fit.
Cut 4–6 small slits in the top to vent steam.
Brush and bake: Brush the crust with beaten egg. Place the dish on a rimmed baking sheet to catch drips and bake for 30–40 minutes, until the crust is golden and the filling is bubbling around the edges.
Rest before serving: Let the pot pie rest for 10 minutes so the filling sets slightly. Slice and serve warm.