01 - Preheat oven to 400°F (200°C). Grease or line a standard 12-cup muffin tin.
02 - Heat vegetable oil in a skillet over medium heat. Add diced onion and cook until translucent.
03 - Add ground beef or pork to the skillet, breaking up with a spoon. Sauté until evenly browned.
04 - Stir in carrot and potato. Sauté for 3 to 4 minutes until beginning to soften.
05 - Add water, Japanese curry roux, soy sauce, granulated sugar, salt, and pepper. Bring to a simmer and cook until vegetables are tender and mixture thickens, about 8 to 10 minutes. Remove from heat and allow filling to cool slightly.
06 - Cut each thawed puff pastry sheet into six squares to yield twelve pieces. Gently press one square into each muffin cup, ensuring the corners remain upright.
07 - Spoon approximately 2 tablespoons of cooled curry filling into the center of each pastry-lined muffin cup.
08 - Fold pastry corners loosely over the filling. Brush exposed pastry surfaces with beaten egg.
09 - Transfer muffin tin to preheated oven. Bake for 18 to 22 minutes, or until pastry is golden and crisp.
10 - Remove muffins from oven and let stand in tin for 5 minutes. Transfer to a wire rack. Serve warm.