Rinse the rice until the water runs clear, then cook it in a rice cooker or on the stovetop at a 1:1.1 rice to water ratio. Keep it hot. This runs in parallel with everything below. For background umami, add a 2-inch strip of kombu to the cooking water and pull it out before serving.

If you bought pre-sliced hotpot, shabu-shabu, or sukiyaki beef, thaw it just enough to separate the sheets. Working from a block, freeze it 1 to 2 hours until firm but not solid, then slice as thin as you can against the grain.

Bring 7 ½ cups of water to a boil in a large pot, turn off the heat, and stir in 1 ½ cups of cold water. Lower in 3 to 4 eggs straight from the refrigerator, cover the pot, and wait 12 minutes. Move them to cold water right away so they stop cooking.

Treat the first batch as a calibration run and pull one egg at 10 to 11 minutes to check, since pot, egg count, and water volume all shift the timing. Keep the 5 to 1 ratio of boiling to cold water when you scale, so two eggs get 5 cups boiling plus 1 cup cold. To use a raw egg at serving instead, skip this step and buy pasteurized in-shell eggs.

Wipe the kombu with a damp cloth and leave the white powder in place. Drop it into 1 liter of cold water in the saucier over medium-low heat. The moment small bubbles start rising, just before it simmers, pull the kombu out. Do not let it boil.

Turn off the heat and add the katsuobushi. Let it steep 2 minutes. Do not stir, which clouds the broth.

Once the flakes have sunk, pour the dashi through a fine mesh strainer. Do not press the flakes, which turns the stock bitter, and do not let them sit past 4 minutes, which turns it fishy. This recipe uses ¾ cup. Cool the rest and refrigerate it covered for up to 3 days.

For the shortcut version, stir Hondashi granules into hot water following the package.

Set a wide, heavy pan or saucier over medium to medium-low heat. Add the oil and the onions, spread in a single layer. Cook, stirring every couple of minutes, until they soften and turn a deep, even golden brown, about 12 to 18 minutes. This is deeper than golden edges but short of full caramelization, and it drives the onion's water off before the sauce goes in.

Pour in the sake and mirin. Simmer at moderate heat until the sharp alcohol smell is gone and the liquid reduces by about half, about 2 to 3 minutes. Reducing them alone strips more raw alcohol than adding everything at once, and the mirin sugars concentrate into a second gentle browning.

Add the dashi and soy sauce, then the sugar and grated ginger. Stir until the sugar dissolves and bring it to a gentle simmer.

Add the sliced beef, spreading the pieces so they separate in the liquid. Simmer gently just until the beef loses its pink, about 1 to 2 minutes. Do not sear it and do not over-simmer, because thin beef toughens and dries when it cooks too long. Skim off the foam and any oil that pools on the surface.

Simmer uncovered and resist the urge to stir. Spoon the sauce over the beef every minute or so, which bastes the slices and keeps them from shredding. Let it reduce until the sauce coats the beef and onions and pools over the rice without flooding the bowl, about 4 to 6 minutes. Taste it before you call it done, since reducing concentrates salt.

Spoon hot rice into each bowl, then the beef, onions, and sauce over the top, adding extra sauce if you like a wetter bowl. Top each bowl with an egg, then finish with beni shoga, scallions, shichimi togarashi, and toasted sesame seeds.