Heat oven to 425°F. In 1-quart saucepan, melt butter over medium-high heat. Add broccoli and milk; cook 2 to 3 minutes, stirring occasionally, just until broccoli is heated. Stir in chicken. Remove from heat.
Separate dough into 4 rectangles; firmly press perforations to seal. Place 1/4 cup broccoli-chicken mixture and 1/4 cup cheese on one half of each dough rectangle. Fold other half of dough rectangle over filling. With fork, seal edges. Gently transfer hand pies to ungreased cookie sheet.
In small bowl, beat egg with water until frothy. Brush tops of hand pies with thin layer egg wash.
Bake 10 to 12 minutes or just until medium golden brown. Cool slightly before serving, 3 to 5 minutes.