Peel and dice the vegetables and set aside.
Brown the ground beef in a large skillet or Dutch oven. Use a colander to drain the fat and set aside.
In the same large skillet where you browned the meat, add the oil and butter on medium-high heat. Once the butter melts, add the carrots and onions to the skillet. Sprinkle with salt and pepper. Stir occasionally just until they begin to soften.
Add the garlic, seasonings, ground beef, tomato paste and Worcestershire sauce stirring to combine. You want the mixture to be nice and hot but not quite boiling. Then sprinkle the flour into the beef mixture and stir to combine. Add the vegetable or beef stock and potatoes then bring to a boil while stirring.
Reduce heat to medium and cover with the lid and allow to simmer until the potatoes are tender, about 25 to 30 minutes. The sauce will thicken while simmering. Stir occasionally.
It is time to remove the puff pastry sheets from the freezer and allow them to thaw at room temperature. If you thawed them earlier, simply keep them in the refrigerator until ready to use.
Once your meat and veggie pie filling is done, remove from heat and allow it to cool before filling the puff pastries.
Preheat the oven to 400°.Cover a counter work space with disposable counter liners for easy cleanup. Place one sheet of puff pastry onto the counter and roll out to about a 10 x 10 inch size. Cut each into fourths so that you have four squares for your hand pies. Continue until you have cut all of the puff pastry. You will have sixteen squares so that you have tops and bottoms for eight hand pies. Line a large baking sheet with parchment paper. Place four pastries onto the sheet pan to be the bottom of the hand pies. Scoop about 1/2 cup of the beef filling onto the center of each pastry.
Place a pastry on top of each pie and press the edges shut with the tines of a fork to seal. Use the fork to poke vent holes in the top of each hand pie. Now place the ready to bake hand pies into the freezer for about 15 minutes before baking. This is because we want the pastry to be firm for baking and they will become soft during the filling process.
Whisk the egg in a small bowl with one teaspoon of water. Remove the hand pies from the freezer. Brush the tops of each hand pie with the egg wash and bake for about 20 to 25 minutes or until golden brown. Cool and enjoy!