Preheat the oven to 400℉. Line a baking sheet with parchment paper or a silicone baking mat.
In a large pot, melt butter. Whisk in flour and cook 1 minute. Slowly whisk in chicken broth until fully combined.
Add garlic powder, onion powder, thyme, salt, and black pepper. Cook for 2-3 minutes until it has thickened to a gravy consistency.
Add shredded chicken and frozen vegetables.
Roll out the puff pastry sheets and cut 6 rectangles from each sheet to make 12 in total.
Add 2-3 tablespoons of filling to the center of 6 rectangles. Place the remaining rectangles on top of the filled sheets. Use a fork to crimp the edges.
Cut small slits in the tops of each hand pie to allow steam to escape. In a small bowl, scramble egg and water. Brush the tops of each hand pie with egg wash.
Bake in the oven for 22-25 minutes until the tops are golden and the bottoms are no longer doughy. Sometimes the edges curl up as they bake; if so, press the corners down.