Put a large pot of salted water on to boil for the ramen. Discard the seasoning packet or reserve for another use. Cook the ramen until tender, drain and rinse with cool water to eliminate any excess starch.
Divide the ramen between 4 serving bowls. Place an egg half and a portion of enoki mushrooms (if using) on top of each nest of noodles. Set aside.
Place the chicken breasts between two sheets of plastic wrap and gently pound them to an even thickness of 1/2-inch. Season on both sides with salt and pepper.
Using the same pot you cooked the ramen in, add the sesame and vegetable oils and heat over medium-high heat.
Working in batches if necessary, add the chicken in a single layer and cook until lightly golden and cooked through, 3 minutes per side.
Transfer to a cutting board and set aside to cool.
Reduce the heat on the pot to medium and add the shallots and ginger.
Sauté until soft and fragrant, 2 minutes, then add the garlic and continue cooking, stirring constantly, until the mixture fragrant, about 2 minutes longer.
Deglaze the pan with the soy sauce and mirin, scraping up any browned bits that may have accumulated on the bottom. Add the chicken broth and bring the mixture to a rapid simmer.
Add the bok choy leaves a few at a time and cook until wilted, about 2 minutes. Using tongs, remove the leaves from the broth and divide between the serving bowls.
Bring the broth back to a rapid simmer and cook for 2 more minutes.
While the broth simmers, slice the chicken breasts crosswise into 1/2-inch strips and transfer to the ramen bowls.
Add the scallions to the simmering broth, remove from the heat immediately and ladle the hot broth over each ramen bowl.
How to Make Medium-Cooked Eggs:
Gently place the eggs in a saucepan large enough to hold them in one layer. Add cool water to the pan to cover them by about 1 inch.
Bring the water to a boil, then immediately remove the pan from the heat, cover it and let stand for 8 minutes (for large eggs).
Drain and cool by running cold water into the pan until the eggs are cool enough to handle for peeling.