In a large bowl, combine the dashi, light soy sauce, mirin, and salt; stir until the salt dissolves. In a medium bowl, use chopsticks or a fork to lightly beat the eggs; add them to the dashi mixture, stirring briefly to combine. (You may still see some streaks of whites; do not overmix.) Divide the egg mixture into four equal portions.
Place a bamboo rolling mat on a cutting board and set aside. Soak a folded paper towel in vegetable oil.
Make the first tamago: Place an 8- by 8-inch tamago pan over medium heat. When the pan is hot, sweep the oiled paper towel over the bottom and sides of the pan. Drop a teaspoon of the egg mixture in the pan to test the temperature; if it sizzles a little, the pan is ready. Ladle about a third of one portion of the egg mixture into the pan, stir for a few seconds with the chopsticks to scramble, then swirl gently to coat the bottom of the pan. Cook over medium heat until the surface of the omelet begins to set (45-60 seconds), then, starting at the edge of the pan furthest from you, use chopsticks or a spatula to fold the sheet of egg toward you into quarters. (Don’t worry too much if your first folding is not perfect—it won’t show.)
Once fully folded, push the omelet to the far end of the pan and use the oily paper towel to re-oil the exposed surface of the pan. Ladle in another third of the egg mixture and use the chopsticks or spatula to gently lift the folded omelet, allowing the egg mixture to get under it and to cover the entire surface of the pan before settling the rolled portion back onto the raw egg. When the second layer of omelet begins to set (45-60 seconds), fold the rolled portion towards you so that the original roll is at the center. Shape the omelet by gently pressing it against the sides of the pan. Repeat the process once more: re-oil the exposed part of the pan, add the remaining third of the egg mixture, cook, and fold.
Transfer the omelet from the pan onto the bamboo rolling mat. Wrap the omelet with the rolling mat, gently pressing to shape it into a rectangle (a traditional shape) while it is warm. Keep the omelet wrapped in the rolling mat while you repeat the whole process to make the remaining 3 rolls. (Rolls will be served at room temperature.)
When ready to serve, in a small bowl, combine the grated radish with the soy sauce. Slice each omelet crosswise into about ½-inch slices. Arrange the omelets on individual serving plates, and serve each with about 1 tablespoon of the soy-radish mixture on the side.