Rinse and soak shitake mushrooms in a bowl of warm water for about 30 minutes to soften. Remove stalk and slice cap thinly. Reserve soaking liquid. Rehydrate wakame by soaking in a bowl of cold water for about 15 minutes. Drain.
Bring a big pot of water to boil. Add soba, reduce heat, and stir frequently to prevent noodles from sticking. If it looks like the water is going to boil over, add ¼ cup (60ml) water to lower the temperature. Soba should be done in 5 to 6 minutes. Remove and drain. Divide soba into 4 bowls.
To make broth, bring 6 cups (1.4 liters) water (including reserved mushroom soaking liquid) to boil in a medium sized pot. Reduce heat, add soy sauce, mirin, dashi, sugar, mushrooms, and carrots. Allow soup to simmer for 2 to 3 minutes. Taste broth and add salt if required. Add wakame. When broth comes back to a boil, turn off heat.
Pour broth over bowls of soba. Garnish with sliced green onions, toasted sesame seeds, shichimi togarashi, and katsuobushi. Serve immediately.