Put 1 tablespoon vegetable oil in a medium saucepan over medium heat. When the oil is hot, add 1 tablespoon sesame seeds, 1½ cups rice, and a sprinkle of salt. Cook, stirring frequently until the rice is glossy and starting to color slightly, 3 to 5 minutes. Add 3 cups of water. Bring to a boil, then adjust the heat so the mixture bubbles steadily but not vigorously. Cover and cook, undisturbed, until small craters appear on the surface, 10 to 15 minutes.
Put 2 tablespoons vegetable oil in a large skillet over low heat. Trim, core, and quarter the cabbage. Cut each quarter crosswise into thin ribbons.
Turn the heat up to medium-high. Add the cabbage to the skillet and sprinkle with salt. Cook, stirring occasionally until the cabbage is lightly browned and wilted (it should still have a little crunch), 6 to 8 minutes.
Crack the eggs into a bowl, sprinkle with salt, and beat them. Trim and chop the scallions.
Move the cabbage to one side of the skillet and add a drizzle of oil to the empty space if it looks dry. Add the eggs and cook, stirring constantly until they’re scrambled and set, a minute or two. Toss the eggs and cabbage together in the skillet and turn off the heat.
When small craters appear on the surface of the rice, tip the pot to see if any liquid remains. If so, cover and keep cooking until the rice is dry, checking every minute or two.
Add the cooked rice to the skillet with the cabbage and eggs. Add the scallions and stir to combine. Taste and adjust the seasoning and serve, passing soy sauce at the table.