Bring a medium sized pot of water to a rapid boil. While you're waiting, heat 1 Tbsp oil in a large sauté pan over medium-high heat. Add the mushrooms and brown evenly, about 5 minutes. Remove the mushrooms to a bowl and turn the heat off.
When the water has begun to boil, add the noodles and stir continuously with a fork or chopstick for 30 seconds. Continue cooking on high for 30 more seconds. Strain in a colander and rinse with cold water for 30 seconds. Set aside.
In the same sauté pan used for the mushrooms, heat the remaining oil over high heat. Add the cabbage and carrots and a pinch of salt. With a wooden spoon, stir and shake the pan constantly, until the vegetables are tender, about 3 minutes. Add the noodles, garlic and eggs. Let cook for 3 minutes longer, stirring only once. In a small bowl, whisk the miso into the soy sauce and add to the noodles. Stir until coated. Divide among 2 shallow bowls. Garnish with the scallion and sprinkle with togarashi.