Bring a small pot of water to a boil. Add the egg and simmer for 6.5 minutes for a jammy yolk. Remove, place in an ice bath, peel, and set aside.
Cook the brown rice ramen noodles in boiling water according to package directions, usually about 4 minutes. Drain and set aside.
Heat the toasted sesame oil in a large skillet or wok over medium-high heat. Add the ground chicken, sea salt, and black pepper, breaking it up with a spatula until browned and cooked through.
Add the sliced mushrooms and carrots to the skillet, sautéing for 3 minutes until they begin to soften.
Stir in the minced garlic and grated ginger, cooking for 30 seconds until aromatic.
Add the chopped bok choy and tamari to the skillet. Toss frequently for 2 minutes until the bok choy leaves are wilted but the stems remain slightly crisp.
Add the cooked ramen noodles to the skillet and toss everything together until the noodles are well-coated and heated through.
Divide the ramen mixture into a bowl. Slice the soft-boiled egg in half and place on top. Garnish with sliced scallions before serving.