Season the chicken with salt and pepper, then place it in a large saucepan and cover with water. Bring to a gentle simmer and cook until just done. Transfer the chicken to a plate and let cool slightly before dicing. Measure and reserve 1¾ cups of the cooking liquid; discard the rest.

In the same saucepan, melt the butter with the celery and onion, sautéing until the vegetables become soft and translucent. Blend in the flour, salt, black pepper, celery seed, garlic powder, and bouillon paste. Gradually stir in the reserved broth along with the milk, cooking over medium-low heat until thickened.

Incorporate the chopped chicken and frozen vegetables into the sauce. Adjust seasoning as necessary. Let the filling cool to room temperature to prevent the crust from softening.

Preheat the oven to 425°F. On a lightly floured surface, roll out one portion of the pie dough into a 12-inch circle and place it into a 9-inch pie dish.

Pour the cooled filling into the crust-lined pan. Roll out the second dough and lay it over the filling. Seal and crimp the edges. Cut a small slit in the top to vent steam.

Whisk the egg with 1 tablespoon milk and lightly brush over the crust to achieve a glossy finish.

Bake the pie for 40 to 50 minutes, until the crust is golden and the filling is bubbling. Shield the crust with foil if it browns too quickly.

Let the pot pie rest for 15 to 20 minutes before serving to allow the filling to set.