Prepare the Filling:
In a large skillet, melt butter over medium heat. Add the onions, carrots, celery, and garlic. Cook for about 5-7 minutes, until softened. Stir in the flour and cook for 1-2 minutes to create a roux.
Make the Sauce:
Gradually add the chicken broth while stirring, then pour in the heavy cream. Stir until the sauce thickens, about 5 minutes. Add the cooked chicken, peas, thyme, salt, and pepper, and cook for an additional 2-3 minutes. Remove from heat and let cool slightly while you prepare the puff pastry.
Prepare the Puff Pastry:
Preheat the oven to 400°F (200°C). Roll out the puff pastry sheets on a lightly floured surface if necessary. Place one sheet in the bottom of your pie dish, pressing gently along the edges.
Assemble the Pot Pie:
Pour the chicken filling into the prepared crust, spreading it out evenly. Lay the second puff pastry sheet over the top of the filling. Trim the excess dough and press the edges to seal the pie. Cut a few slits in the top crust to allow steam to escape.
Bake:
Brush the top of the pie with the beaten egg to create a golden, glossy finish. Bake for 25-30 minutes, or until the puff pastry is golden brown and crispy.
Serve:
Let the pot pie cool for a few minutes before serving. Slice into portions and enjoy!