Rinse the rice and slice the ginger into thin matchsticks.
Put the rice, broth, water and ginger in a large heavy pot. Bring to a boil over high heat, then lower to low, partly cover and simmer for 50 minutes, stirring every 10 minutes so it does not stick, until the rice breaks down into a thick porridge.
Add the whole chicken breasts for the last 15 minutes and simmer until they reach 74°C (165°F). Lift out, shred and stir back in.
Meanwhile, bring a small pot of water to a boil over high heat, lower in the eggs and boil for 7 minutes. Cool in ice water, peel and halve.
Ladle the congee into bowls and top with the eggs, sliced green onion, soy sauce and sesame oil.
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