Pat chicken dry with paper towels and season with salt and pepper. Heat oil in large Dutch oven over medium-high heat until just smoking. Cook chicken until browned, about 10 minutes; transfer to plate. Pour fat (you should have 2 tablespoons; supplement with butter if necessary) into bowl; reserve. When chicken is cool enough to handle, remove and discard skin. Add broth, chicken, and bay leaf to now-empty pot and bring to boil. Reduce heat to low and simmer, covered, until breasts register 160 degrees and thigh register 175 degrees, 14 to 18 minutes. Transfer chicken to bowl. When chicken is cool enough to handle, shred into bite-size pieces, discarding bones. Pour broth through fine-mesh strainer into second bowl and reserve (you should have about 2¾ cups); discard bay leaf
Adjust oven rack to lowest position and heat oven to 450 degrees. Heat butter and reserved fat in now-empty pot over medium heat until shimmering. Add flour and cook, whisking constantly, until golden, 1-2 minutes. Slowly whisk in 2 cups reserved broth and half-and-half and bring to boil. Reduce heat to medium-low and simmer gravy until thickened and reduced to 1¾ cups, 6-8 minutes. Season with salt and pepper to taste. Combine 1 cup gravy with shredded chicken; reserve remaining gravy for serving.
Transfer chicken mixture to dough-lined pie plate and spread into an even layer. Top with second dough round, leaving at least ½-inch overhang all around. Fold dough under so that edge of fold is flush with rim of pie plate. Flute edges using thumb and forefinger or press with tines of fork to seal. Cut four 1-inch slits on top. Brush pie with egg and bake until top is light golden brown, 18-20 minutes. Reduce oven temperature to 375 degrees and continue to bake until crust is deep golden brown, 10-15 minutes. Let pie cool on wire rack for at least 45 minutes.
When ready to serve, bring remaining ¾ cup of reserved gravy and remaining ¾ cup reserved broth to boil in medium saucepan. Simmer over medium-low heat until slightly thickened, 5-7 minutes. Season with salt and pepper to taste. Serve pie with gravy.