FOR THE CRUST: Combine sour cream & egg in bowl. Process flour & salt in food processor until combined, about 3 seconds. Add butter & pulse until only pea sized pieces remain, about 10 pulses. Add 1/2 of sour cream mixture & pulse until combined, 5 pulses. Add remaining sour cream mixture & pulse until dough begins to form, about 10 pulses.
Transfer mixture to lightly floured counter & knead briefly until dough comes together. Divide dough in 1/2 & form each 1/2 into 4" disk. Wrap each disk in plastic wrap & refrigerate for at least 1 hour or up to 2 days.
(Dough can be wrapped tightly in plastic & foil & frozen for up to 2 months. Thaw completely at room temperature before using.)
Remove 1 dough disk from refrigerator & let sit for 10 minutes. Working on lightly floured counter, roll into 12" round & transfer to 9" pie plate, leaving 1/2" overhang all around. Repeat with 2nd dough disk & transfer to parchment lined rimmed baking sheet. Cover both doughs with plastic & refrigerate for 30 minutes.
FOR THE PIE: Pat chicken dry with paper towels & season with salt & pepper. Heat oil in large dutch oven over medium-high heat until just smoking. Cook chicken until browned all over, about 10 minutes; transfer to plate. Pour fat (you should have 2 Tbsp) into small bowl; reserve. When chicken is cool enough to handle, remove & discard skin. Add broth, chicken & bay leaf to now-empty pot & bring to boil. Reduce heat to low & simmer, covered, until breasts register 160 degrees F & thighs register 175 degrees F, 14 - 18 minutes. Transfer chicken to bowl. When chicken is cool enough to handle, shred into bite sized pieces, discarding bones. Pour broth through fine mesh strainer into 2nd bowl & reserve (you should have about 2 3/4 C); discard bay leaf.
Adjust oven rack to lowest position & heat oven to 450 degrees F. Heat butter & reserved fat in now-empty pot over medium heat until shimmering. Add flour & cook, whisking constantly, until golden, 1 - 2 minutes. Slowly whisk in 2 C of the reserved broth & 1/2 & 1/2 & bring to boil. Reduce heat to medium low & simmer gravy until thickened & reduced to 1 3/4 C, 6 - 8 minutes. Season with salt & peppeer to taste. Combine 1 C gravy with shredded chicken; reserve remaining gravy for serving.
Transfer chicken mixture to dough lined pie plate & spread into even layer. Top with 2nd dough round, leaving at least 1/2" overhang all around. Fold dough under itself so that edge of fold is flush with outer rim of pie plate. Flute edges using thumb & forefinger or press with tines of fork to seal. Cut 4 1" slits in top. Brush pie with egg & bake until top is light golden brown, 18 - 20 minutes. Reduce oven temperature to 375 degrees F. & 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 3/4 C reserved gravy & remaining 3/4 C reserved broth to boil in medium saucepan. Simmer over medium low heat until slightly thickened, 5 - 7 minutes. Season with salt & pepper to taste. Serve pie with gravy.