Lower the eggs into boiling water for 6 minutes 30 seconds, then straight into iced water. Peel once cool and halve just before serving.
Bring the dashi to a bare simmer in a pot. Slide in the whole chicken breast, cover, and poach at the gentlest possible bubble for 12 to 14 minutes until it reaches 65C, or until just cooked through. Rest it on a board for 5 minutes, then slice.
While the chicken rests, add the tofu and wakame to the broth and simmer 2 minutes.
Turn the heat off. Ladle a cup of broth into a bowl, whisk the miso into it until smooth, then stir it back into the pot. Miso that boils loses most of its aroma, so this order matters.
Taste and adjust with soy sauce. Divide the tofu and broth between bowls, fan the sliced chicken on top, add an egg half or two per bowl, and finish with scallions.