Pulse the toasted sesame seeds in a food processor until they are finely ground. Add the garlic, mayonnaise, miso, lemon juice, sugar, and water, then process until the sauce is smooth and creamy.

Bring a large pot of salted water to a boil and cook the udon until tender. Drain the noodles well and divide them among individual serving bowls.

Combine the water, kombu, wakame, mirin, fish sauce substitute, and sugar in a large pot or Dutch oven. Bring the mixture to a simmer over medium-high heat and cook for 5 minutes. Use a slotted spoon to remove and discard the kombu once the broth is fragrant.

Stir the carrots into the simmering broth and cook until they just start to soften, about 3 minutes. Add the bok choy, tofu, and mushrooms, and continue to simmer until the vegetables are tender and the tofu is heated through, about 3 to 5 minutes.

Ladle the hot broth and vegetables over the bowls of udon. Serve hot, topped with a generous dollop of the sesame sauce.