To make the broth, first combine the water, dried shiitakes, and kombu in a small saucepan. Place it over medium heat and bring the water to a simmer. Immediately remove it from the heat, then use a fine-mesh strainer to remove the dried shiitakes and kombu (Note 2).
Stir the soy sauce, mirin, and sugar into the strained liquid. Set the mixture aside.
Carefully remove the tofu from the packaging, gently blot it with paper towels, then cut the block into eight cubes.
Generously coat the bottom of a large nonstick skillet with oil, pouring it to about ¼-inch deep. Place the skillet over medium heat.
While the oil heats up, place the cornstarch into a small bowl.
Once the oil begins to shimmer, carefully coat a piece of the tofu in the cornstarch, shaking off any excess, then transfer it to the hot oil. Repeat for as many pieces of tofu as you can fit into the skillet without crowding. Fry the tofu cubes for about 3 to 4 minutes, then gently turn the pieces over and fry for another 3 to 4 minutes, until golden brown and crispy. Transfer them to a wire rack to drain when done, then repeat the process for any remaining tofu.
To serve, warm the broth back up if it has cooled down, divide the broth among small bowls, then place a few tofu cubes in each bowl. Top with grated daikon, ginger, and chopped scallions. Serve.