Bring a medium saucepan of water to a boil. Gently lower in the eggs and boil for exactly 6-7 minutes for perfectly soft yolks. Immediately transfer eggs to an ice bath to stop cooking. When cool, peel and set aside.
In a large pot, heat the sesame oil over medium heat. Add minced garlic and ginger paste, sautéing until fragrant—about 1-2 minutes. Stir carefully to prevent burning.
Add the sliced mushrooms to the pot and cook until they soften and release their juices, roughly 5 minutes. Pour in the chicken stock and water, bringing the mixture to a gentle simmer.
Stir in soy sauce, then add salt and pepper to taste. Let the broth simmer for 10-15 minutes to deepen the flavors.
In a separate pot, boil water and cook ramen noodles according to package instructions, usually 3-4 minutes. Drain and set aside.
Drop the baby bok choy into the simmering broth for the last 2-3 minutes. Add the sliced grilled chicken to warm through.
Divide noodles between serving bowls. Ladle hot broth, mushrooms, bok choy, and chicken over the noodles. Top each bowl with halved soft-boiled eggs, sliced green onions, and a sprinkle of sesame seeds.