In a large bowl, whisk together flour and salt. Add diced, chilled butter. Use a pastry blender or your fingertips to cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining.

Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Be careful not to overmix. Form the dough into a disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes.

While the dough chills, prepare the filling. Heat olive oil in a large skillet over medium heat. Add ground beef (or chicken) and cook, breaking it up with a spoon, until browned. Drain any excess fat.

Add diced onion, carrots, and potatoes to the skillet. Cook for 5-7 minutes, until vegetables start to soften. Stir in dried thyme, garlic powder, and black pepper.

Sprinkle 1 tablespoon of flour over the meat and vegetable mixture, stir for 1 minute to cook the flour. Pour in the beef (or chicken) broth, stirring constantly until the mixture thickens slightly. Remove from heat and let cool completely. Stir in the shredded cheddar cheese.

Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.

On a lightly floured surface, roll out the chilled dough to about 1/8-inch thickness. Using a 4-5 inch round cookie cutter or a bowl, cut out circles of dough. Reroll scraps as needed.

Spoon about 2-3 tablespoons of cooled filling onto one half of each dough circle, leaving a small border.

Moisten the edges of the dough with a little water, then fold the other half of the dough over the filling to create a half-moon shape. Crimp the edges with a fork to seal securely.

Place the hand pies on the prepared baking sheet. Cut a small slit or two on the top of each pie to allow steam to escape. Brush the tops with the beaten egg wash.

Bake for 20-25 minutes, or until the hand pies are golden brown and puffed. Let cool slightly on a wire rack before serving.