In a large bowl, combine the dried hijiki with plenty of water and let it soak for about 30-60 minutes. (If the package specifies a soaking time, follow those instructions.) Once the hijiki is rehydrated, drain it in a fine-mesh strainer, rinse it under running water, and drain well.
Cut the carrot into thin strips. Pat the aburaage dry with paper towels to remove any excess oil, then cut it into thin strips.
Heat oil in a pot. Once the oil is hot, add the carrot and stir-fry briefly over medium heat until evenly coated with oil. Add the hijiki and continue cooking for about 1-2 minutes to remove any excess moisture. Then, add the aburaage and soybeans, and stir-fry briefly to combine.
Add the seasonings (dashi, mirin, sugar, and soy sauce) to the pot and bring to a boil. Once boiling, reduce the heat to medium-low and simmer for 10-15 minutes, stirring occasionally, until only a small amount of liquid remains.
Remove from the heat and let the mixture cool in the pot, or transfer it to a bowl along with any remaining liquid and let it cool, allowing the flavors to meld. It can be served as is, but chilling it in the refrigerator for a while longer will allow the flavors to develop further.