Make the crust: Stir sour cream and egg. Pulse flour and salt in a food processor; add butter until pea-sized. Add sour cream mixture in two parts, pulsing until dough forms.
Chill & roll: Divide dough into two disks, wrap, and chill 1 hour. Roll one into pie plate, second onto baking sheet. Chill 30 minutes.
Cook chicken: Season and brown in oil. Simmer with broth and bay leaf until cooked (14–18 min). Shred meat; strain broth.
Make gravy: Melt butter with reserved fat. Whisk in flour, then broth and half-and-half. Simmer until thickened. Season to taste. Mix 1 cup gravy with chicken.
Assemble pie: Fill crust with chicken mixture, top with second crust, seal, cut slits, and brush with egg.
Bake: 450°F for 18–20 min, then 375°F for 10–15 min, until golden. Cool 45 min.
Serve: Warm remaining gravy with extra broth; serve alongside slices.