Boil the eggs for 6 minutes, then transfer to an ice bath. Peel and set aside.

Toss the chicken with soy sauce and 1 teaspoon sesame oil. Sear in a hot skillet with neutral oil for 5 minutes per side until cooked through, then rest and slice against the grain.

In the same skillet over medium heat, sauté garlic and ginger for 30 seconds. Add broth and mushrooms, scraping up browned bits, and simmer 5 minutes.

Whisk miso with a ladle of hot broth, then stir into the pot with rice vinegar and remaining 1 teaspoon sesame oil. Remove from heat.

Cook noodles according to package directions, drain, and rinse briefly.

Divide noodles among bowls, ladle broth over, and top with sliced chicken, halved eggs, scallions, sesame seeds, and chili oil.