In a large mixing bowl, combine bread flour, cake flour, and salt. In a separate bowl, whisk together warm milk, unsalted butter, granulated sugar, and instant yeast. Combine both mixtures and knead the dough until smooth and elastic—about 10-12 minutes by hand or 8-10 minutes with a mixer.

Transfer the kneaded dough to an oiled bowl, cover it with a clean towel, and let it rise in a warm spot for 60-90 minutes, or until it has doubled in size.

Heat a frying pan over medium heat and sauté chopped onion, carrot, and potato until tender. Next, add ground beef and cook until browned. Stir in soy sauce, ketchup, and the broken-down Japanese curry roux, then mix in a cornstarch slurry until the filling thickens.

After the dough has risen, punch it down and knead briefly. Divide the dough into 16 equal pieces, rolling each into a ball. Flatten the balls, add a spoonful of curry filling, seal the edges carefully, and coat each piece with egg and breadcrumbs.

Arrange the assembled curry bread on a baking sheet, cover it lightly, and let them rise again for 20-30 minutes.

Heat vegetable oil in a deep pan to 350°F (180°C). Fry the bread in batches until golden brown, about 4-5 minutes. Drain on paper towels to remove excess oil.