Step 1: 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.
Step 2: 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.
Step 3: In small bowl, beat egg with water until frothy. Brush tops of hand pies with thin layer egg wash.
Step 4: Bake 10-12 minutes or just until medium golden brown. Cool slightly before serving. 3-5 minutes.