Pour the soy sauce, mirin, and sugar into a skillet and bring to a boil. Reduce to a simmer and let cook for 5 minutes. Transfer to a resealable container.
Sauté onion and sauté until just softened. Stir in the mushrooms and cook until lightly browned. Pour in a ¼ cup of the kaeshi, ½ a cup of the dashi, and the sugar.
Let cook for 15-20 minutes or until most of the moisture has been absorbed. Transfer the mushrooms to a bowl and set aside.
Pour ½ a cup of the kaeshi, and the mirin into the simmering dashi and stir. Drain the noodles and rinse in cold water. Divide the noodles into four bowls and pour the broth over top.
Top with a couple of spoonfuls the mushroom mixture and add a quarter of a sheet of nori to each bowl. Garnish with scallions and furikake and serve immediately.