Place eggs in a saucepan and cover with water.

Bring to a boil, remove from heat, and let eggs stand in hot water for 2 minutes.

Prepare a bowl of ice water while eggs cook.

Transfer eggs from the saucepan to the ice bath, reserving cooking water in the saucepan.

Soak until completely cooled, about 15 minutes.

Gently peel eggs and set aside.

Heat sesame oil and olive oil together in a skillet over medium-high heat.

Add garlic and ginger paste and sauté 1 minute, stirring constantly.

Add vegetable broth and bring to a simmer.

Add soy sauce and mirin.

Add mushrooms to the simmering broth and cook 5 minutes.

Add bell peppers, bok choy, and green onions and cook 1 minute more.

Reheat the water used for the eggs.

Add ramen to the boiling water and cook until just tender, 2 to 3 minutes.

Drain the ramen noodles.

Place a portion of cooked noodles in a bowl and top with broth, vegetables, and 1 egg.

Garnish with kimchi, chile-garlic sauce, and radish.

Repeat with remaining noodles, broth, vegetables, egg, and toppings.