Make dashi. Add 4 cups water and kombu in a medium pot. Cover and bring to a boil over high heat. Reduce heat to medium-low and simmer for 10 minutes. Remove from heat, add bonito flakes, and cover. Let sit for 10 minutes. Strain, discarding kombu and bonito flakes. Set aside.
Prepare shoyu tare. Combine all of the ingredients for tare in a medium saucepot. Bring to a boil then reduce heat to low and simmer for 10 minutes. Strain and set aside to cool.
Marinade soft boiled peeled eggs in 1 tablespoon shoyu for at least one hour or overnight. Turn eggs halfway through to allow marinade to coat all sides. Cut eggs in half.
Cook rice. Add rice with dashi, shoyu, and chashu if using, in a medium pot. Bring to a boil over high heat. Cover and reduce heat to low, and cook for 15 minutes or until rice is tender. Remove from heat and let sit for 5 minutes. Remove cover and stir rice. Let cool slightly before handling.
Assemble egg onigiri. Use a 2 ½ inch biscuit cutter and fill the center with rice. Scoop out a small amount of rice in the center, large enough to fit ½ an egg. Lather the center with about 1 teaspoon kewpie and lay egg half on top. Carefully remove biscuit cutter, leaving rice intact. Wrap onigiri with a thin nori strip. Repeat with remaining ingredients. Serve warm.