Heat a large pot over medium high heat and add the sesame oil and olive oil.
Add the julienned carrots, sliced crimini mushrooms, and the chopped green onions and saute in the oil until slightly softened.
While the vegetables are cooking, set a medium pot of water to boil. Once it reaches a boil, add the eggs and cook until soft-boiled (about 7 minutes). Remove the eggs from the pot and run them under cool water to stop the cooking. Set aside.
Once the vegetables are starting to become soft add the ginger and soy sauce, cooking for about 1 minute.
Add the vegetable stock and stir well to combine.
Once the mixture reaches a simmer, drop the dried ramen noodles into the pot and stir well as they begin to loosen. Cook the noodles for about 2 or 2 and a half minutes over medium heat.
As the noodles cook, peel the boiled eggs.
Add the seaweed to the pot and stir gently.
Ladle the soup into bowls (about 2 cups per serving) and top each one with a soft-boiled egg cut in half.
Sprinkle some sesame seeds and chopped green onions over each bowl, and serve.