Whisk all the sesame sauce ingredients together in a small bowl until the mixture is smooth and completely combined.

Bring 2 quarts of water to a boil in a large pot. Add the udon and salt, stirring frequently to prevent the noodles from sticking together, and cook until tender. Drain the noodles and set them aside.

Bring the water, kombu, and wakame to a brief boil in a large pot over medium heat. Remove from the heat and use a slotted spoon to discard the seaweed once it has infused the liquid.

Stir in the mirin, fish sauce, and sugar and return the broth to a simmer over medium heat. Add the bok choy and carrots and simmer until they are crisp-tender, about 2 to 4 minutes.

Gently stir in the tofu, mushrooms, and cooked noodles. Let the soup cook for about 1 minute until everything is heated through. Divide into bowls and drizzle each portion with the prepared sesame sauce before serving.