In a large skillet, cook the beef, onions, garlic, thyme, Worcestershire sauce, tomato paste, salt, and pepper over medium heat until the beef is browned and the vegetables are softened, about 10 minutes.
Add the potatoes and cook for another 10 minutes, stirring occasionally.
In a small saucepan, melt the butter over medium heat. Add the flour and stir to combine. Cook for 1 minute, stirring constantly.
Gradually add the beef broth, whisking until smooth. Cook until the mixture thickens, about 5 minutes.
Add the beef and potato mixture to the saucepan and stir to combine. Cook for 2 minutes, stirring occasionally.
Preheat the oven to 375°F (190°C). Line a baking sheet with parchment paper.
Unfold the puff pastry sheets and cut into 8 equal rectangles. Place about 1/4 cup of the beef and potato mixture in the center of each rectangle. Brush the edges of the pastry with the beaten egg.
Fold the pastry over the filling and press the edges together to seal. Transfer the pasties to the prepared baking sheet.
Brush the tops of the pasties with the remaining beaten egg. Bake for 20 minutes, until golden brown.
Interesting Facts
Pasties have been a popular dish in England since the 13th century.
The traditional Cornish pasty is made with shortcrust pastry and filled with beef, potatoes, onions, and turnips.
The pasty was a staple of the miner's lunch, as the crust would not crumble in their pockets.