Adjust an oven rack to the middle position and heat the oven to 400 degrees. Cover a rimmed baking sheet with foil.
Pat the chicken dry with paper towels and season with salt and pepper. Place on the prepared baking sheet, skin-side up, and roast 35 to 40 minutes. Let cool to room temperature, remove and discard the skin, and cut the meat from the bone and shred the chicken into 2-inch pieces. Reduce the oven temperature to 350 degrees.
Melt the butter in a large saucepan over medium heat. Cook the celery until softened, about 5 minutes. Add the garlic and cook until fragrant, about 30 seconds. Stir in the flour and cook until it begins to brown, about 1 minute. Stir in the milk, broth, and zest and bring to a boil. Reduce the heat to low and simmer until thickened, about 5 minutes. Strain the sauce into a large bowl, discarding the vegetables. Stir in the chicken and scallions, and season to taste with salt and pepper. Let the filling cool until just warm, about 30 minutes.
Spoon the chicken mixture into the pie shell and sprinkle with the cheese. Top with the remaining circle of dough. Trim all but 1/2 inch of the dough overhanging the edge of the pie, using your fingers.
Cut four 2-inch slits in the top of the dough, brush with the beaten egg, and bake until the crust is deep golden brown, 45 to 55 minutes. Let cool 10 minutes. Serve.