Wipe the kombu gently with a damp paper towel, add it to the pot with 4 cups cold water, and soak for 15 to 30 minutes. Heat over medium until tiny bubbles form around the edges, about 5 to 7 minutes, then remove the kombu just before boiling.
Add the bonito flakes, turn off the heat, and steep for 3 to 5 minutes until the flakes sink. Strain through a fine-mesh strainer into a bowl, discard the flakes, and rinse out the pot; you should have about 3½ cups of golden dashi.
Pat the chicken dry and cut into ¾-inch pieces. Remove shiitake stems and slice caps ¼ inch thick, cut carrots into thin half-moons, and separate bok choy stems from leaves. Drain and cube the tofu, rehydrate the wakame in cool water for 5 minutes then drain and squeeze, and slice scallions keeping whites and greens separate.
Heat the neutral oil in the clean pot over medium heat until it shimmers. Add the scallion whites, ginger, and garlic and stir for about 30 seconds until fragrant, being careful not to brown the garlic.
Add the sliced shiitake mushrooms and carrots and cook for 4 to 5 minutes, stirring often, until the mushrooms release their liquid and begin to soften.
Pour in the strained dashi and chicken broth, scraping up any bits stuck to the bottom. Stir in the soy sauce and mirin, raise the heat, and bring to a gentle boil.
Add the chicken pieces, lower the heat so the broth simmers softly with small bubbles, and simmer for 8 to 10 minutes until the chicken is white throughout and reaches 165°F. Skim off any foam that rises to the surface.
Stir in the chopped bok choy stems and simmer for 2 minutes. Add the bok choy leaves and cubed tofu and simmer for 1 to 2 minutes more, then turn the heat to low so the soup is steaming but no longer bubbling.
Ladle about 1 cup of hot broth into a small bowl, add the white miso paste, and whisk or muddle until completely smooth with no lumps. Turn off the heat under the pot, pour the miso mixture back in, and stir gently to combine.
Stir in the rehydrated wakame and toasted sesame oil. Ladle into four bowls, dividing the chicken, vegetables, and tofu evenly, and sprinkle the scallion greens over the top. Serve right away while hot and fragrant.