Bring a small pot of water to a boil. Carefully lower the egg into the water and boil for 6 minutes and 30 seconds. Immediately transfer to an ice bath, then peel and set aside.
In a medium saucepan, heat the sesame oil over medium heat. Add the minced garlic and grated ginger, sautéing for 1 minute until fragrant.
Add the sliced chicken breast to the saucepan. Cook for 5-7 minutes until the chicken is browned and cooked through.
Pour the chicken bone broth into the pan and bring to a gentle simmer.
Place the miso paste in a small bowl and stir in two tablespoons of the warm broth until smooth. Pour the mixture back into the saucepan.
Add the high-protein noodles to the simmering broth and cook according to package instructions, typically 3 to 5 minutes.
During the final 2 minutes of noodle cooking, add the bok choy and edamame to the saucepan to blanch.
Pour the broth and noodles into a deep bowl. Slice the soft-boiled egg in half and place it on top of the noodles.
Garnish with sliced green onions and serve immediately.