01 - Set oven temperature to 400 degrees Fahrenheit.
02 - In a large skillet, melt the butter over medium heat. Add onions, garlic, carrots, and celery; cook until softened, 5 to 7 minutes.
03 - Sprinkle the flour over the vegetables, stirring continuously for 1 to 2 minutes to eliminate the raw taste.
04 - Gradually whisk in chicken broth and milk; continue stirring until mixture is smooth and thickened, about 4 to 5 minutes.
05 - Add chicken, peas, thyme, parsley, salt, and pepper. Stir to combine and allow to simmer for 2 to 3 minutes, then remove from heat.
06 - Transfer the filling into a 9-inch round pie dish or equivalent baking vessel.
07 - Place puff pastry or pie crust over the filling, trim excess, and crimp edges to seal. Cut small slits in the crust to vent steam.
08 - Brush the crust evenly with beaten egg to achieve a golden finish.
09 - Bake for 25 to 30 minutes, or until the crust is golden and filling bubbles.
10 - Allow to rest for 10 minutes before serving.