Pour in the broth, then add the mushrooms and chicken. Bring to a gentle simmer over medium heat, about 6 minutes, until tiny bubbles rise around the edge.

Reduce the heat to low, set the lid slightly ajar, and simmer for 20 minutes, until the thickest thigh reads 165°F and the meat looks firm and pale with clear juices. A hard boil makes the thighs rubbery and the broth cloudy.

Boil 2 quarts of water in the saucepan, lower in the cold eggs, and cook for 7 minutes, until the water keeps a lively bubble around each egg.

Move the eggs to ice water for 5 minutes, then peel and halve them, until the whites look set and the yolks look jammy and orange.

Whisk the soy sauce, miso, sesame oil, rice vinegar, and sugar until smooth and glossy with no lumps. Divide it among four bowls, about 1 1/2 tablespoons each.

Move the chicken to a cutting board and shred it with two forks, until it pulls apart easily with no pink at the center.

Pour the broth through the strainer into a large heatproof bowl, then return it to the pot over low heat. Trim the woody shiitake stems and slice the caps thin, until the broth looks clear and golden.

Refill the saucepan with fresh water and boil it. Cook the noodles for 3 to 4 minutes (the package time minus 30 seconds), adding the bok choy for the last minute, then drain well, until the noodles look glossy with a slight chew and the bok choy turns bright green.

Ladle about 1 3/4 cups of hot broth into each bowl and stir until the tare dissolves. Add the noodles, chicken, bok choy, mushrooms, eggs, green onion greens, and sesame seeds. Serve within 5 minutes, while the yolks still glow orange.