Place chicken in a large skillet over medium-high heat. Cover and cook, stirring halfway, until no longer pink in the center, 6 to 12 minutes.
Pour half the chicken broth into the skillet with the chicken. Add frozen vegetables, dehydrated onion, granulated garlic, minced garlic, gravy mix, and salt. Stir and cover; reduce heat to medium and cook until flavors meld, about 8 minutes. Add instant potatoes and stir until incorporated. Add remaining chicken broth and stir. Cover and reduce heat to low; cook until thickened to your liking, about 8 minutes more.
Remove skillet from heat and allow to cool, about 30 minutes. Pour into a container, cover, and refrigerate filling, 2 hours to overnight.
Preheat the oven to 325 degrees F (165 degrees C). Spray 2 baking sheets with cooking spray.
Open and unroll dough into one continuous sheet. Place on a prepared baking sheet. Pinch all seams together, sealing them. Stretch the dough to twice its size using a small rolling pin. Score the sheet into 8 squares using a knife. Repeat with the remaining dough on the other baking sheet.
Place 1 heaping tablespoon of filling into the center of a dough square. Pull the side up and over the mixture. Seal the edge carefully with your fingers and then using a fork. Repeat with remaining dough and filling.
Bake in the preheated oven, watching carefully until just starting to brown, about 13 minutes. Brush pies with beaten egg and cook until dried, about 2 minutes more. Allow to cool for a few minutes before eating.