Heat a large pot or Dutch oven over medium heat. Add the sesame oil. Once it shimmers, add the minced garlic and ginger. Stir constantly for about 30-60 seconds until fragrant. Be careful not to let the garlic burn, as it will become bitter.
Add the sliced mushrooms to the pot. Cook, stirring occasionally, for 3-5 minutes. The mushrooms will soften, brown slightly, and release their natural juices, adding a deep, savory (umami) flavor to the broth.
Pour the chicken stock, water, and soy sauce into the pot. Stir everything together and bring it to a boil. Once boiling, reduce the heat to low and let it simmer gently for at least 10 minutes. This allows all the flavors to meld together. For an even richer flavor, you can add a tablespoon of mirin.
While the broth is simmering, bring a separate, large pot of water to a boil. Do not cook the noodles in your ramen broth. Cooking them separately prevents the broth from becoming starchy and cloudy, and it ensures your noodles have the perfect texture. Cook the ramen noodles according to the package directions (usually 2-4 minutes). Once cooked, drain them well.
In the last 2 minutes of the broth simmering, add the baby bok choy. Cook just until the leaves are tender but still bright green. Do not overcook.
Divide the drained noodles evenly among your serving bowls. Arrange the sliced grilled chicken on top of the noodles. Ladle the hot broth, along with the mushrooms and bok choy, over the noodles and chicken in each bowl.
Top each bowl with a halved soft-boiled egg, a generous sprinkle of sliced green onions, and a dash of sesame seeds. Serve immediately while hot!