Preheat the oven to 400°F with a rack in the center position.
Melt the butter in a large skillet over medium heat. Add the onion, carrots, and celery and cook, stirring, until the onion is tender and translucent, about 5 minutes. Add the garlic and cook, stirring, until fragrant, about 1 minute more. Add the flour, salt, pepper, celery seed, and thyme and cook, stirring, until a thick paste forms, about 2 minutes.
Gradually stir in the milk until fully incorporated. Once the milk is fully incorporated, gradually stir in the stock. Simmer, stirring continuously, until the mixture has thickened, about 3 minutes. Stir in the peas and chicken until combined. Remove from the heat.
Line a 9-inch pie pan with one of the pie crusts. Spoon the filling into the pie crust, forming a slight mound in the center. Brush the edge of the crust with some of the beaten egg, then cover the filling with the remaining pie crust. Using a fork, crimp the edges of the bottom and top crusts together to seal. Using a sharp knife, cut two 2-inch slits into the top of the crust. Brush the top of the crust with the remaining beaten egg.
Place the pie plate on a rimmed sheet pan. Bake until the crust is lightly browned and the filling is bubbling, 45 to 50 minutes. Cool on a baking rack for 10 to 15 minutes before serving.
