Prepare the Filling: In a large pan, melt the dairy-free butter over medium heat.
Add the finely diced onion, diced carrots, and diced celery. Cook for about 5-7 minutes, until the vegetables are softened.
Add the minced garlic and dried thyme. Stir and cook for another 1-2 minutes until fragrant.
Sprinkle the plain flour over the vegetables. Stir well to coat the veggies and cook for 1-2 minutes to eliminate the raw flour taste.
Gradually pour in the dairy-free milk, stirring constantly to avoid lumps. Continue to stir until the mixture thickens, about 3-4 minutes.
Add the shredded cooked chicken and frozen peas to the pan. Stir everything together and cook for another 2-3 minutes until heated through. Season with salt and pepper to taste. Remove the filling from the heat and let it cool slightly.
Assemble the Pie: Preheat your oven to 200°C (400°F).
Spoon the chicken and vegetable filling into the pie dish. Place the puff pastry sheet(s) over the filling, trimming any excess pastry around the edges.
Press the edges of the pastry down with a fork to seal the pie. Make a few small slits in the top of the pastry to allow steam to escape. Brush the beaten egg over the top of the pastry.
Bake the pie for 25-30 minutes, or until the pastry is golden brown and crispy.
Once baked, remove the pie from the oven and let it cool for a few minutes before serving.