In a bowl mix together soy sauce, hoisin sauce, water and ginger and whisk to combine. Remove one tablespoon sauce for cooking with the tofu.
In a wok or large or pan over high heat add oil and let warm for 1-2 minutes, or until oil is very hot. Add in mushrooms, broccoli and carrots and let cook for 3 minutes, then add in 1/3 of the sauce and cook for 2 more minutes.
Add in the Japanese udon noodles and another 1/3 of the sauce, cook for 5 minutes, stirring/tossing until the noodles are coated in sauce, pour remaining 1/3 of the sauce and turn heat off. If the noodles and vegetables are sticking to the side of the pan, add a few tablespoons water to make the dish more saucy.
While the noodles are cooking, make the tofu.
Pat tofu cubes dry with a paper towel. Toss tofu cubes with corn starch in a bowl, sprinkle with a pinch of salt.
Over high heat, heat oil, then add tofu cubes and 1 tablespoon sauce. Let cubes cook for 1-2 minutes per side so that a nice brown crust forms. Remove from heat when all sides are browned. Add tofu to the noodles in the wok and toss everything so that everything is combined.
To assemble the Japanese noodle bowls divide the contents of the wok into 4 bowls and top with bean sprouts, green onions and black sesame seeds (if using), then enjoy!