Heat a medium skillet over medium-high heat, add the butter, and let it melt. Add the onion and cook, stirring frequently, until the onion is translucent and light brown at the edges, about 8 minutes. Let the onions cool. Meanwhile, combine the panko and milk in a medium bowl, stir to moisten the crumbs, and let it sit for 5 minutes or so.
Add the onions, panko mixture, beef, salt, nutmeg, and white pepper to the bowl with the panko mixture and mix firmly with your hands until the ingredients are well distributed and the mixture is slightly sticky to the touch, about 1 minute. Add the egg and mix again, about 30 seconds more.
Using lightly oiled hands, grab about a quarter of the meat mixture and firmly toss it back and forth between your hands for about 1 minute, or 30 seconds if you’re quick. (The goal—unlike that of most Western hamburger makers—is to get rid of most of the air hiding out in the patties.) Form 4 patties of more or less equal size (about 4 1⁄2 inches in diameter and 1⁄2 inch thick). Use your fingers to make a gently sloping dent in the center of each patty; this way, they won’t puff too much during cooking.
Heat the oil in a large skillet over medium-high heat until it begins to smoke. Add the patties, leaving a little space between each one, and cook until the undersides are deep golden brown, 2 to 3 minutes. Lower the heat to medium, flip over the patties, and cook until the undersides are browned, about 3 minutes more.
Meanwhile, stir together the teriyaki sauce, ketchup, and mustard in a small bowl and add the mixture to the skillet. Let the sauce come to a gentle simmer and cook, flipping the patties occasionally, until they are cooked to medium or medium well, 3 to 5 minutes. Serve right away.