01 - Heat the oven to 400°F.
02 - Melt butter with olive oil in a large skillet over medium heat. Add diced onion and minced garlic, cooking 2 to 3 minutes until softened and fragrant.
03 - Add diced carrots and celery to the skillet and cook for 5 minutes, stirring occasionally until they begin to soften.
04 - Stir in frozen peas, corn, and cooked chicken. Season with thyme, sage, salt, and black pepper; mix thoroughly.
05 - Sprinkle all-purpose flour evenly over the mixture and cook, stirring constantly, for 2 minutes to develop a roux.
06 - Gradually pour in chicken broth while stirring to prevent lumps. Add milk or heavy cream, bring to a gentle simmer, and cook for 5 to 7 minutes until the sauce thickens.
07 - Remove from heat, taste filling, and adjust salt or pepper if necessary.
08 - Transfer the filling to a 9-inch pie dish or an equivalent baking vessel.
09 - Roll out puff pastry if needed to cover dish. Place pastry over the filling, trim excess, and crimp edges. Cut small vents on top to release steam.
10 - Brush the pastry surface evenly with the beaten egg.
11 - Bake for 25 to 30 minutes until the crust turns golden brown and the filling bubbles.
12 - Let the dish cool for 10 minutes before slicing and serving.