Bring a large pot of water to a boil. Add the noodles and cook as the label directs. Remove the noodles to a colander using tongs and rinse under cold water; toss with the sesame oil and set aside. Add the spinach to the boiling water; cook until wilted, about 2 minutes. Drain and rinse under cold water; squeeze dry and set aside.
Rinse out the pot; add 6 cups water, the carrots, garlic and ginger. Bring to a simmer over medium-high heat. Cook until the carrots soften, about 8 minutes. Stir in the mushrooms and soy sauce; simmer until the mushrooms are tender, about 2 minutes.
Divide the noodles and spinach among bowls; ladle the broth and vegetables into the bowls. Top each serving with the trout, scallions and chile pepper.