Combine kampyo, 1 tsp. salt, and 3 cups water in a bowl; let sit 1 hour, then drain and transfer to an 8" non-stick skillet. Add 2 tbsp. granulated sugar and ½ tbsp. soy sauce; simmer over medium-high heat until soy sauce has been absorbed, about 5 minutes and transfer to a bowl.
Combine mushrooms and 2 cups boiling water in a bowl; let sit until soft, 30 minutes, then drain and transfer to skillet. Add 1½ tbsp. granulated sugar, 1½ tbsp. soy sauce, the mirin, and 2 tbsp. water; simmer until liquid has been absorbed, 5 minutes. Thinly slice mushrooms and transfer to a bowl.
Return skillet to medium-high heat; add 1 tsp. oil and cook halibut, flipping once, until cooked through, 2–3 minutes. Add 1½ tbsp. granulated sugar, the sake, and salt; reduce heat to medium-low and cook, stirring to break fish up into small flakes, until liquid has been absorbed, 8-10 minutes. Transfer fish to a bowl.
Cook mizuna and carrots and in salted boiling water until tender, about 30 seconds for mizuna and 2 minutes for carrots; transfer to a bowl of ice water, then drain and pat dry using paper towels.
Wipe skillet clean; add kombu and ¼ cup water. Simmer over medium heat for 2 minutes; remove from heat and stir in bonito flakes. Let cool, then strain into a bowl. Add remaining granulated sugar, the eggs, and salt; whisk to combine. Heat skillet over medium heat; cook ¼ egg mixture, stirring occasionally until just set, 1–2 minutes. Using a rubber spatula, roll eggs onto one side of the pan. Add another ¼ eggs; cook until just set and starting from initial egg roll, fold and roll eggs and push to side of pan. Repeat with remaining eggs; transfer 1 large roll to a cutting board and slice lengthwise into 3 strips.
Bring rice and 2¼ cups water to a boil in a 2-qt. saucepan; reduce heat to low and cook, covered, until rice is tender, 15–20 minutes. Remove from heat and transfer to a bowl; immediately stir in vinegar, confectioners’ sugar, and salt; fan rice while stirring until rice is cool enough to handle.
Assemble maki rolls: Working with 1 sheet nori at a time, place nori on sushi mat, shiny side down. Using wet fingers, press ⅓ rice onto nori in an even layer, leaving a ½" border on the side furthest from you. Working from the side closest to you, arrange 2 pieces kampyo and ⅓ each fish, mushrooms, egg mixture, mizuna, and carrots. Using the mat as a guide, lift edge of mat closest to you to roll nori up and over ingredients, forming a tight cylinder. Take care to pull back the edge of the sushi mat so as to not roll it into the sushi. Transfer sushi roll to a cutting board and slice crosswise into 8 pieces; serve with soy sauce, and if you like, pickled ginger and wasabi paste.