Wash the potatoes under cold running water. Peel the potatoes with a vegetable peeler and remove the eyes.
Cut each potato into equal 4 pieces. Tip: To cook evenly, it's important that they are of similar size.
Put the water and potatoes in a large pot, cover, and bring to a boil over medium heat, leaving the lid slightly ajar to prevent the pot from boiling over. It will take about 15 minutes to come to a boil. Reduce the heat to maintain a gentle boil and cook until done, about 15-20 minutes.
Prepare the rest of the ingredients while you wait for the potatoes to cook. First, mince the onion finely: Cut the onion in half from root end to tip. Lay one half flat on the cutting board. With the knife tip pointing toward the root end, make ¼-inch vertical slices to within ½ inch of the root end. Next, make ¼-inch horizontal slices, keeping the root end intact.
Then, cut perpendicular down through the first slices you made. If the onion pieces need to be chopped finer, you can run your knife through them in a rocking motion. Be sure to hold down the tip of the knife; otherwise, the onions are going to go flying around the room. Repeat with the other half.
In a large skillet, heat the oil on medium heat and add the chopped onions.
Sauté the onions until translucent and tender, about 12-15 minutes. Tip: Make sure there is no moisture left in the onions so that the croquette don't get soggy. Leave the chopped onions for a long stretch between stirring to give them a nice char. Take care not to burn the onions.
Add the meat to the pan and break it up with a wooden spoon.
Season the mixture with salt and black pepper.
Stir to combine and cook until the meat is no longer pink. Turn off the heat and remove the pan from the stove to let it cool slightly.
Once in a while, check if the potatoes are done cooking. Insert a skewer into a bigger piece of potato; if it pierces the potato easily, it's done. Remove the potatoes from the heat and drain off the water completely. When you do so, use a lid to keep the potatoes from falling out of the pot.
Move the pot back to the stove. Shake the pot over low heat and let the remaining moisture completely evaporate (without burning the potatoes), about 2-3 minutes. Then, transfer them to a large bowl.
With a potato masher, mash the potatoes while they are still hot, allowing the steam to escape in the process. Add the butter. Tip: Keep some potato chunks for texture.
Add the salt and pepper and combine well.
Gently squeeze the liquid from the meat mixture by collecting it to one side of the pan. We do this to avoid adding excess moisture to the potatoes.
Add the drained meat mixture to the bowl with the mashed potatoes and mix to combine.
To Form the Patties
While the mixture is still warm (but not hot), form it into oval-shaped patties roughly 3 inches in length. Take care not to create air pockets in the patties.
Placed the patties on a tray or plate. Cover and let the Korokke patties rest in the refrigerator for 15-30 minutes. Do not skip this step!Tip: Cooling the patties prevents the croquettes from exploding while deep-frying. If the patties are cold, they will not create and release steam. If you skip this process, the internal temperature of the warm patties will rise in the hot oil and create steam, which will then puncture a hole in the panko coating and explode. Resting in the refrigerator also helps the ingredients meld together.
To Bread the Croquettes
Prepare one tray each for the flour and the panko. Crack the eggs into a small bowl and beat with a fork.
Dredge each patty in flour and shake off the excess.
Then, dip each patty in the egg and coat with the panko, pressing the breadcrumbs to make sure they adhere.
Once you finish breading the patties, add the oil to a medium pot. Make sure there's at least 2 inches (5 cm) of oil in the pot so the croquettes are completely submerged.
Bring the oil to 340-350ºF (171-178ºC) over medium heat. Use a cooking thermometer to check the oil temperature (so there's no guessing!) Once you get used to deep-frying, you can use the chopstick or panko tricks to determine if the oil is ready for deep-frying.
Deep-fry 2-3 croquettes at a time until they are golden brown, about 2-3 minutes. Tip: Do not touch the croquettes until one side is golden brown. The inside is already cooked, so all you need to do is to fry until they're golden brown!
Transfer the croquettes to a wire rack or a plate lined with paper towels to drain the excess oil. Continue deep-frying the rest of the patties.
Serve the croquettes with tonkatsu sauce (here's my homemade recipe). Typically, deep-fried foods are served with shredded cabbage in Japan.
You can store the leftovers in an airtight container and freeze up to a month. To reheat, put the defrosted or frozen croquettes on a baking sheet lined with aluminum foil or parchment paper. Bake at 350ºF (180ºC) for 15-20 minutes if defrosted or 45 minutes if frozen. Check if the inside is warm before serving.