Melt the butter in a skillet over medium heat and sauté the onion, carrot, and celery until softened. Stir in the flour and cook for about a minute to remove the raw taste, then slowly whisk in the chicken broth and milk until the mixture thickens into a creamy sauce.

Add the shredded chicken, peas, thyme, and a pinch of salt and pepper. Stir until everything is well combined, then remove from the heat and let it cool completely. Chilling the filling for at least 30 minutes makes assembling much easier.

Roll out the puff pastry on a lightly floured surface and cut it into six rectangles or circles (6 in each pastry sheet), roughly 5 inches wide. Spoon a small amount of the chilled filling into the center of half of the pieces, leaving a border around the edges. Brush the edges lightly with the egg wash and place the remaining pastry pieces on top. Press around the edges to seal and crimp with a fork. Cut small vents on top to let steam escape.

Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper. Place the pies on the sheet, brush the tops with more egg wash, and bake for 25 to 30 minutes, or until golden and puffed. Let them cool for a few minutes before serving; the filling will thicken slightly as they rest.