Boil the potatoes until they are soft enough to mash. Drain and mash them thoroughly.
In a pan, sauté the chopped onions, carrots, and ground meat until they are softened and cooked through.
In a separate saucepan, melt butter over medium heat.
Add flour and stir well to form a roux. Gradually add milk, stirring constantly to avoid lumps. Cook until the sauce thickens. Season with salt and pepper.
Mix the mashed potatoes, sautéed vegetables/meat, and white sauce together. Adjust seasoning if necessary.
Preheat your oven to 375°F (190°C).
Shape the mixture into oval or round patties.
Dip each korokke into beaten eggs and then coat them with breadcrumbs.
Place the coated beef croquettes Japanese on a baking sheet lined with parchment paper. Lightly brush them with a bit of oil.
Bake in the preheated oven for about 25-30 minutes, or until they are golden brown and crispy.
Allow the korokke to cool for a few minutes before serving.