Rinse the dried daikon briefly, then cover it with cold water in a large bowl and soak for about 15 minutes, until it swells to roughly three times its volume and turns soft but still chewy.
Drain and squeeze the daikon firmly with both hands to remove as much water as possible. Any water left behind will dilute the broth. Cut very long strands into shorter lengths.
Peel the carrot and cut it into matchsticks about the same thickness as the daikon strands. Cut the aburaage into thin strips.
Heat the oil in a frying pan over medium heat. Add the daikon, carrot, and aburaage together and stir-fry for 2 to 3 minutes, until the carrot softens. Do not skip this - coating everything in oil keeps the daikon from going mushy.
Add the water, dashi powder, soy sauce, sugar, and mirin. Bring to a simmer over medium heat and cook uncovered, stirring occasionally, for 10 to 15 minutes, until the liquid has reduced to about a third.
Take the pan off the heat while a little glossy broth still pools in the pan - the daikon keeps absorbing as it cools. Serve warm, at room temperature, or cold.
