Preheat oven to 350 degrees. Divide dough in half. On a floured sheet of parchment paper, roll out one half to a 14-inch round. With a knife or biscuit cutter, cut out six 4 1/4-inch circles (rerolling dough once if necessary) and transfer, on parchment, to a baking sheet. Repeat with remaining dough, cutting out six (larger) 4 1/2-inch rounds. Chill rounds on sheet until ready to use.
In a large skillet, melt butter over medium-high. Add leek and cook, stirring, until soft, 3 minutes. Add kale and thyme, season with salt and pepper, and cook until kale wilts, 3 minutes. Sprinkle flour over mixture and stir to combine. Add broth and bring to a boil. Cook, stirring often, until mixture thickens, 2 minutes. Transfer to a medium bowl, season with salt and pepper, and stir in chicken. Let cool slightly.
Place a heaping 1/4 cup chicken mixture on each of the smaller dough rounds, leaving a 1/2-inch border. Brush edges with egg and top with larger dough rounds; using fingers, press edges firmly to seal. Cut a small vent in each pie. Bake until browned and crisp, 30 minutes, rotating sheet halfway through. Let cool slightly on sheets on a wire rack. Serve warm or at room temperature.
If using store-bought rolled pie dough, add 1 tablespoon cheddar inside each hand pie.