Cook and Mash the Potatoes: Cut the peeled potatoes into large chunks. Boil in salted water until incredibly fork-tender. Drain completely. While still steaming hot, transfer to a large bowl and mash them until mostly smooth. Let them cool slightly.

Cook the Meat: Melt the butter in a skillet over medium heat. Sauté the minced onions until very soft, sweet, and golden. Add the ground beef, cooking until perfectly browned. Season aggressively with salt and black pepper.

Mix the Base: Add the savory, oily beef and onion mixture directly into the mashed potatoes. Fold everything together. Divide the mixture into 4 or 6 equal portions and shape them into thick, oval patties. Let them chill in the fridge for 20 minutes to firm up.

Bread the Korokke: Set up a breading station. Dredge each potato patty completely in flour, dip it entirely into the beaten egg, and press it firmly into the Panko breadcrumbs to create a thick, dense crust.

Fry: Heat 1.5 inches of vegetable oil in a deep skillet to 340°F (170°C). Carefully lower the breaded patties into the oil. Fry for 2 to 3 minutes per side until deeply golden brown and shatteringly crisp. Drain on a wire rack and serve piping hot with Tonkatsu sauce.