In a medium pot, bring 8 cups of water to a low boil and carefully add 6 eggs. Gently simmer for 12 minutes. Remove the eggs from the pot and set aside to cool. (Keep the water in the pot so you can use it to cook the noodles.) Once the eggs are cool enough to handle, remove the peel and cut each egg in half. Set aside.
In a skillet over medium low heat, cook the ginger in the olive oil for a minute. Then add the red pepper and mushrooms to the skillet. Cover and saute for 3 minutes, stirring occasionally. Then add the sliced cabbage and cook for another minute. Add the broth and soy sauce, heat briefly, then remove the skillet from the stove and set aside.
Meanwhile, return the medium pot from the eggs to the stove and bring to a boil. Cook the noodles according to the package directions (about 3 minutes), then drain. Add the noodles to the skillet with the vegetables. Stir to combine.
Divide the ramen among 6 bowls. Top with carrots, scallions, and hard boiled eggs, and serve warm.