Season the chicken breast with salt and pepper. Sear in a pan over medium-high heat until the internal temperature reaches 165°F (74°C). Rest for 5 minutes, then slice into thin strips.
Bring a small saucepan of water to a boil. Lower the egg into the water and simmer for 6 minutes and 30 seconds. Immediately transfer to an ice bath for 3 minutes, then peel and halve.
In a medium pot, heat sesame oil over medium heat. Sauté the garlic and ginger for 60 seconds until fragrant.
Pour in the chicken bone broth and soy sauce. Bring the liquid to a gentle simmer.
In a small ramekin, whisk the miso paste with 2 tablespoons of the warm broth until fully emulsified, then stir the mixture back into the main pot.
Add the baby bok choy and shelled edamame to the broth. Simmer for 3 minutes until the vegetables are tender-crisp.
Submerge the high-protein ramen noodles into the broth and cook for the duration specified on the package (typically 2-4 minutes) until al dente.
Transfer the noodles and broth to a large bowl. Arrange the sliced chicken, soft-boiled egg, and green onions on top before serving.