Thaw the puff pastry in the refrigerator according to the package directions, keeping it cold until ready to use.

Brown the ground beef in a large skillet over medium-high heat for 5–7 minutes, breaking it into small pieces. Drain excess fat if necessary.

Add the diced onion and potato, then cook for 6–8 minutes, stirring occasionally, until the onion softens and the potato begins to become tender.

Stir in the garlic, tomato paste, 1/2 teaspoon of the kosher salt, black pepper, and thyme. Cook for about 1 minute.

Sprinkle in the flour and stir for 30–60 seconds. Add the beef broth and Worcestershire sauce, then simmer for 2–4 minutes until thick and the potato is tender.

Remove the skillet from the heat and cool the filling for at least 10–15 minutes. Stir in the shredded Cheddar once the mixture has cooled slightly.

Heat the oven to 400°F and line a baking sheet with parchment paper. Cut each puff pastry sheet into four equal squares, making eight squares total.

Place approximately 1/3 cup of filling in the center of each pastry square. Fold each square into a triangle or rectangle, then press and crimp the edges firmly with a fork.

Transfer the pockets to the prepared baking sheet and keep them cold while the oven finishes heating. Beat the egg with the water and brush the tops, then cut one or two small vents in each pocket.

Bake at 400°F for 20–25 minutes, until the pastry is puffed and deeply golden brown. Rest the pockets for 5 minutes before serving.