Cut the potatoes into 2 to 6 pieces depending on their size, and place them in a pot with the skin on. Add enough water to fully cover the potatoes and enough salt to make a 1% salt solution (about 1.6 tsp of salt per 1 quart or 1 liter of water; this salt is not included in the ingredient list). Bring to a boil, then reduce the heat to low and simmer until the potatoes are tender, about 15-20 minutes (they should be soft enough that a skewer or fork slides in easily).
(You can do this step while the potatoes are boiling.) Finely chop the onion. Heat butter in a pan, and once it has melted, add the onion and sauté over medium heat until softened. Then, add the ground beef and pork, season with salt and pepper, and cook until the meat is no longer pink.
Drain the potatoes in a colander. While they are still warm (and being careful not to burn yourself), peel off the skins by hand. Transfer the warm potatoes to a bowl and mash them until there are no large lumps remaining.
Add the onion and ground meat mixture, along with sugar, to the bowl of mashed potatoes, and mix until well combined and smooth. Once the mixture has cooled to room temperature, cover the bowl with plastic wrap and chill in the refrigerator for at least 30 minutes.
Divide the chilled mixture into equal portions according to the number of korokke you want to make (I divided mine into eight). Shape each portion into an oval or round patty about ¾ inch (2 cm) thick. (In Japan, a typical patty weighs about 2.7 ounces / 70-80 g.)
Crack the egg into a small bowl and beat it. Prepare two containers, one with flour and the other with panko breadcrumbs.
Coat each patty in this order: flour, beaten egg, and panko breadcrumbs. Make sure the patty is thoroughly coated at each stage.
Heat oil in a pot to 340℉ (170℃). Once the oil is hot, carefully add the patties one at a time. Deep-fry them for about 5-6 minutes until they turn golden brown, flipping them occasionally.Be careful not to add too many patties at once, as this can cause the oil temperature to drop significantly. The patties should occupy no more than about two-thirds of the oil’s surface area.If you don't have a thermometer, drop a few panko breadcrumbs into the oil to test the temperature. When the breadcrumbs float to the surface at a moderate speed without touching the bottom, it indicates that the oil is ready. Remove the fried patties from the oil and let them drain on a wire rack. If there are any fried bits left in the oil, skim them out with a mesh skimmer or strainer, then repeat the frying process (step 8) until all the patties are fried.
Serve the korokke with fresh vegetables, such as shredded cabbage, if desired. They are delicious as is, but you can also drizzle them with Worcestershire sauce or tonkatsu sauce to taste.