Brown the beef and vegetables. Heat 1 tbsp neutral oil in a heavy pot over medium-high heat. Add the small beef cubes and brown for about 3–4 minutes in batches if needed. Reduce the heat to medium, add the diced onion and carrot, and stir for about 5 minutes. Add the garlic, ginger, and tomato paste and cook for 30–60 seconds, stirring.

Simmer the filling. Pour in the beef broth, scraping up any browned bits. Cover and simmer gently for approximately 75–95 minutes, until the beef is fork-tender; continue simmering longer if needed. Check the pot occasionally and add a little water if the liquid is getting low. The timing depends on the cut and size of the pieces.

Make the dough while the beef simmers. In a large mixing bowl, combine the flour, instant yeast, sugar, and salt. Mix in the warm milk, 1 egg, and 2 tbsp oil. Knead for 8–10 minutes until smooth and elastic, adding only a little extra flour if the dough is excessively sticky. Cover and let rise in a warm place for about 60 minutes, or until noticeably puffy and roughly doubled.

Finish and cool the curry. When the beef is tender, uncover the pot and reduce any excess liquid. Take the pot off the heat and stir in the chopped Japanese curry roux until dissolved; return to low heat for 2–3 minutes, stirring, until the filling is thick and holds together on a spoon. If too thick, add a splash of water; if runny, simmer a little longer. Spread the filling in a shallow bowl and cool until no longer hot before shaping.

Fill and shape the buns. Punch down the risen dough and divide it into 8 equal portions. Roll each portion into an oval about 5 inches (13 cm) long. Spoon about 2–3 tbsp of cooled thick curry onto one side, leaving a clean border. Fold the dough over the filling, firmly pinch the seam closed, and place seam-side down. Do not overfill. Cover the shaped buns loosely while working.

Coat the buns. Beat the remaining egg in a shallow bowl and place the panko in another bowl. Brush or gently dip each bun in egg, then coat all sides with panko, pressing lightly so it adheres. Let the coated buns rest for about 10 minutes while heating the oil.

Fry in small batches. In a heavy, deep pot, heat neutral oil to 325–340°F (165–170°C), monitoring it with a deep-fry thermometer and keeping the pot no more than half full of oil. Fry 1–2 buns at a time, turning gently, for about 4–6 minutes total, until evenly golden brown and the bread is cooked through. The exact time depends on their size. Check the first bun for uncooked dough before proceeding. Transfer to a wire rack to drain.

Serve. Allow the filling to cool briefly before serving. Serve warm for the best contrast between the crisp coating and soft bread.