Preheat the Oven: Preheat oven to 425°F with a rack positioned in the center.
Cook the Vegetables: Melt the butter in a 12-inch cast-iron or oven-safe skillet over medium heat. Add the onion, carrots, and celery and cook, stirring occasionally, until the vegetables are softened and the onion is translucent, about 5 minutes.
Build the Roux: Sprinkle the flour, salt, pepper, celery seed, garlic powder, and thyme evenly over the vegetables. Stir constantly until the mixture forms a thick paste that coats the vegetables, about 2 minutes.
Make the Sauce: Slowly pour in the milk, whisking constantly until smooth. Gradually whisk in the chicken stock. Bring the mixture to a gentle simmer and cook, stirring continuously, until thickened to a gravy-like consistency, about 5 minutes.
Add Chicken and Peas: Stir in the frozen peas and diced chicken until evenly combined and warmed through.
Prepare the Puff Pastry: On a lightly floured surface, roll the puff pastry into a roughly 16-inch square. Cut into 3-inch squares (you should have about 16 pieces).
Top the Pie: Arrange the puff pastry squares over the filling, slightly overlapping to cover most of the surface.
Egg Wash: In a small bowl, beat the egg with 1 tablespoon of water. Brush the tops of the puff pastry with the egg wash.
Bake: Transfer the skillet to the oven and bake for 25 minutes, or until the puff pastry is deep golden brown and the filling is bubbling. If the pastry browns too quickly, loosely cover with foil.
Serve: Remove from the oven and let rest for a few minutes before serving.
Notes
If you don’t have a cast-iron or oven-safe skillet, prepare the filling in any large skillet through Step 5. Transfer the mixture to a greased baking dish, then continue with the puff pastry topping and baking instructions.