Combine the kombu, dried shiitake, and 1.4 L / 6 cups water in a 3-quart / 2.8L saucepan. Let it soak for 10 minutes (or up to overnight in the fridge) to pull out flavor.

Set the pan over medium heat and bring to a gentle simmer, about 8 minutes. Remove the kombu just before the broth starts to bubble, since boiling it makes the broth slimy and bitter.

Let the dried shiitake simmer 5 more minutes, then strain the broth through a fine-mesh strainer into a bowl. Reserve the softened mushrooms, slice thin, and set aside.

Wipe out the saucepan and heat the sesame oil over medium. Add the fresh shiitake and cook for 5 to 6 minutes, stirring occasionally, until browned and the edges look crisp.

Add the garlic and ginger and cook for 30 seconds until fragrant.

Pour the strained dashi back into the pan along with the sliced rehydrated shiitake and soy sauce. Bring to a gentle simmer.

Add the tofu cubes and simmer gently for 3 to 4 minutes, just to warm through. Keep the heat low, a hard boil breaks up the tofu.

Ladle about 1/2 cup of the hot broth into a small bowl and whisk in the miso paste until smooth. Stir this back into the pot off the heat so the miso stays fragrant and doesn't turn bitter.

Stir in the wakame if using and let it sit 2 minutes to soften. Ladle into bowls, top with scallions and a drizzle of sesame oil, and serve right away.