Cut dough into 4 equal portions; cover and let stand 10 minutes.
Combine buttermilk and next 8 ingredients (through garlic).
Steam broccoli 1 minute. Rinse with cold water; drain. Combine broccoli, buttermilk mixture, chicken, and cheese in a large bowl.
Roll each dough portion into a 6 1/2-inch circle. Place about 1 cup filling slightly off-center on each. Fold dough over filling; crimp edges to seal. Place on a parchment-lined baking sheet. Bake at 450° for 10 minutes. Brush with egg; bake at 450° for 10 minutes.