Soft boil the eggs: Fill a medium pot of water and bring it to a boil, then reduce to a simmer. Gently lower the eggs into the water with a spoon and simmer for 7 minutes. Meanwhile, prepare a bowl of ice water. When the eggs are done, place them in the ice bath and let them cool completely, then gently peel.
Meanwhile, thinly slice the garlic. Peel and mince the ginger.
In another large pot or Dutch oven, heat the sesame oil and olive oil over medium high heat. Add the garlic and ginger and saute for 1 minute, stirring constantly. Add the vegetable broth and bring to a simmer. Add the soy sauce and mirin.
Remove the stems from the shiitake mushrooms and thinly slice them. If necessary, wash the greens. Thinly slice the peppers and slice the green onions (cut the green onions on the bias). Thinly slice the radishes for a garnish, and set aside. Place the mushrooms in the broth and cook 5 minutes. Add peppers, green onion and greens and cook for 1 minute. Taste the broth and season with salt until the flavor pops, adding a splash of soy sauce if needed.
In the pot used for the eggs, bring fresh water to a boil and boil the ramen noodles until just tender, around 2 to 3 minutes, then drain.
To serve, place noodles in a bowl and top with broth, vegetables, and soft-boiled eggs. Garnish with radishes and garlic chili sauce.