Finely chop the onion. Heat oil in a pan, add the onion along with salt, and sauté over medium heat until softened and translucent. Once softened, remove the pan from the heat and let the onion cool to room temperature, either in the pan or by transferring it to a separate dish to speed up the cooling process.
In a bowl, combine the ground beef, ground pork, salt, black pepper, and nutmeg. Knead the mixture until it becomes sticky and cohesive. Once it reaches this stage, add the cooled onions, egg, panko breadcrumbs, and milk, and continue kneading until everything is well combined.
Divide the mixture into equal portions according to the number of menchi katsu you want to make (I divided mine into six portions). Take one portion and shape it into a ball, then gently toss it back and forth between your hands about 10 times, as if playing catch, to remove any air pockets. 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 320℉ (160℃). Once the oil is hot, carefully add the patties one at a time. Deep-fry them for 4 minutes, flipping once halfway through.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. Remove the fried patties from the oil and let them rest on a wire rack for at least 4 minutes. If there are any fried bits left in the oil, skim them out with a mesh skimmer or strainer. Repeat steps 6 and 7 until all the patties are fried.
Increase the heat and raise the oil temperature to 360℉ (180℃). Then, carefully add the patties back into the oil one at a time and fry them for about 2 minutes until golden brown. As before, avoid adding too many patties at once.
Remove the 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 steps 8 and 9 until all the patties are double-fried.
Serve the menchi katsu 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.