Make Boiled Eggs: Bring a medium pot of water to a boil over medium heat. Carefully place 2 eggs straight from the fridge into the water, adding them slowly to prevent temperature shock. Boil for 6–7 minutes, then transfer them to an iced water. Once cold, peel the shells (see Note 1).
Prepare the Toppings: Shred ⅛ cabbage and make or reheat 2 servings karaage.
Prepare the Sauce: Add all the ingredients for one type of sauce (your choice!) to a medium pot and heat over medium heat until the sugar melts.
Coat the Karaage: Add the karaage to the pot and simmer until the sauce thickens, stirring occasionally to prevent burning. Turn off the heat when most of the sauce is absorbed, but a little remains in the pot.
Assemble the Don: In a bowl, add cooked Japanese rice, shredded cabbage, and the sauce-coated karaage. Top with a boiled egg, Kewpie mayonnaise, Sesame seeds, and 2 tablespoons minced green onions and 1 teaspoon sesame oil for the sweet and sour version.