Preheat oven to 400 degrees.
Line a rimmed baking sheet with parchment paper; set aside.
In a large bowl, mix together the beef, ketchup, mustard, salt, and pepper. Form mixture into 8 equal-size patties.
In a large skillet, heat the oil over medium-high heat until shimmering. Brown the patties in oil about 1 minute per side.
On a lightly floured surface, until the pastry. Cut the pastry lengthwise into four equal strips, then cut strips in half crosswise.
Place one patty on each strip and top with cheese.
Fold the pastry up and around the burgers and pinch the edge together to seal, molding to fit around the patties.
Combine beaten egg and 1 teaspoon of water. Brush the tops of the pastries with the egg mixture.
Plane the mini burger on the prepared baking sheet. Bake 12 to 16 minutes or until puffed and golden.
Let stand 5 minutes before serving.