Prepare Filling: In a large skillet, heat 1-2 tablespoons of vegetable oil over medium heat. Sauté 1 chopped onion until soft and translucent, about 5 minutes. Add 1 diced carrot and 1 diced potato, cooking for another 5 minutes. Incorporate 250g of ground beef and cook until browned. Pour in 200ml of water and simmer until vegetables are tender, roughly 10–15 minutes. Stir in Japanese curry roux until thickened, and allow to cool completely.

Make Dough: In a mixing bowl, combine 250ml of warm milk, 2 tablespoons of sugar, and 2 teaspoons of instant yeast. Let it foam for about 5 minutes. In another bowl, mix 500g of bread flour and 1 teaspoon of salt. Pour in the milk mixture, 1 beaten egg, and incorporate until a dough forms. Knead for 10 minutes, gradually adding 50g of unsalted butter until smooth and elastic. Place in a greased bowl, cover, and let rise for 1 hour or until doubled.

Shape Buns: After rising, punch down the dough and divide into 10-12 portions. Shape each into a ball, then flatten each ball until about 4 inches wide. Spoon 1-2 tablespoons of cooled curry filling into each, pinching edges to seal. Arrange on a baking sheet, cover, and let rest for an additional 20-30 minutes.

Cook Buns: For frying, heat vegetable oil to 350°F (175°C). Gently lower buns into oil, cooking for 3-4 minutes on each side until golden. For baking, preheat oven to 375°F (190°C). Brush buns with egg wash and bake for 15-18 minutes until golden and puffed.

Serve: Once cooked, let the buns cool slightly on a wire rack. They can be served warm, ideally with a side of miso soup.