Nori Maki (Nori Rolls with Japanese Omelette, Shiitake Mushroom, and Halibut)

Ingredients

• 3 strips kampyo (dried gourd strips), halved
• 1 tsp. kosher salt, plus more to taste
• 7 tbsp. granulated sugar
• 2 tbsp. soy sauce, plus more for serving
• 6 dried shiitake mushrooms, stemmed and hydrated
• 1 tbsp. mirin
• 1 (2-oz.) piece boneless, skinless halibut
• 1 ¹⁄₂ tbsp. sake
• 16 sprigs mizuna, tough ends discarded
• 1 medium carrot, cut into ¼" matchsticks
• 1 ¹⁄₂ cups sushi rice, rinsed until water runs clear
• 2 ¹⁄₂ tbsp. rice vinegar
• 1 tbsp. confectioners’ sugar
• 1 (2") square piece kombu
• 1 tsp. dried bonito flakes
• 2 eggs
• Sushi mat, for rolling
• 3 sheets nori, lightly toasted over a gas burner or using a blowtorch
• Pickled ginger and wasabi paste, for serving (optional)