Preheat oven to 400F (200C).
Cut chicken into 1-inch (2.5-cm) pieces.
Heat a small amount of olive oil in a large frying pan and add onion and celery. Stir often until softened, about 3 to 4 minutes.
Add chicken and frozen vegetables and cook until just warmed. Remove chicken and vegetables to a bowl.
Return pan to burner and reduce heat to medium. Add butter. When butter is melted, gradually whisk in flour until smooth. Cook for about 1 minute.
Slowly whisk in milk. Whisk until the mixture is thickened, 2 to 3 minutes.
Remove the pan from the heat and add salt, thyme, and the chicken and vegetable mixture. Stir until the mixture is evenly coated with the sauce. It will be very thick.
Pour mixture into a 10 x 8-inch baking dish or a dish that will hold 8 cups and place on a rimmed baking sheet.
On a lightly floured surface roll puff pastry out until it's slightly larger than the pan. Lay over the filling pressing edges of pastry onto the rim of the dish. Using a sharp knife, pierce the middle of the pastry in 3 or 4 places to allow steam to escape. Brush the top with the beaten egg.
Bake in the center of a preheated oven until the pastry is golden and the filling is bubbly, approximately 30 to 35 minutes. Let stand for about 10 minutes before serving to allow the sauce to thicken.