01 - Preheat oven to 400°F. Grease or line a 12-cup muffin tin to prevent sticking.
02 - Heat vegetable oil in a skillet over medium heat. Add diced onion and sauté until translucent.
03 - Add ground meat to skillet, breaking up with a spoon. Cook until browned and no longer pink.
04 - Stir in diced carrot and potato. Sauté for 3 to 4 minutes until lightly softened.
05 - Add water, curry roux, soy sauce, sugar, salt, and pepper. Simmer until vegetables are tender and mixture has thickened, approximately 8 to 10 minutes. Remove from heat and allow filling to cool slightly.
06 - Cut each puff pastry sheet into 6 equal squares (total 12 squares). Gently press each square into a muffin cup, allowing corners to extend above the rim.
07 - Spoon cooled curry filling into pastry cups using about 2 tablespoons per muffin.
08 - Fold pastry corners loosely over the filling. Brush exposed pastry with beaten egg for a golden finish.
09 - Bake for 18 to 22 minutes, or until pastry is puffed, golden brown, and crisp.
10 - Let muffin pies cool in tin for 5 minutes, then transfer to wire rack. Serve warm.