Cook the udon noodles according to the package directions, drain and set aside.
In a large saucepan, bring the stock to a boil. Lower the heat to medium and add the carrots and cook until the carrots are crisp-tender, 1 to 2 minutes. Add the snow peas and cook until slightly tender but still bright green, about 1 minute. Add the mushrooms, cook for 30 seconds and remove from the heat.
Spoon the miso in a medium bowl and add a ladleful of hot broth. Whisk until the miso is completely dissolved, then pour the entire miso mixture into the pot with the soup. Do not boil the soup with the miso, as the miso will become gritty. Stir in the noodles and the green onions.