Add the dry hijiki to a bowl and cover with plenty of water. Let it rehydrate for 20 minutes and then drain and rise.
Prepare the enoki mushrooms by trimming off the bottom third and discarding and then cutting the remaining enoki into thirds (about 1 1/4-inch long pieces).
Heat a frying pan over medium-high heat until hot and swirl with oil. Add the julienned carrots, enoki, and sugar and saute until until the mushrooms have released some liquid, and that liquid has evaporated.
Add the aburaage and continue sauteing for another 30 seconds.
Add the hijiki and toss to coat evenly with oil, then add the dashi, sake, soy sauce and salt.
Turn up the heat to high and cook tossing frequently to reduce the liquid and coat the hijiki evenly with sauce. It's done when there is no liquid left in the pan.
Finish by tossing the hijiki together with the sesame seeds.