Bring 5 quarts cold water to a boil in a large pot.
Reduce the heat to a simmer, add the chicken breasts, and gently poach for 15 to 20 minutes, or until cooked through with an internal temperature of 165°F.
Remove the chicken from the poaching liquid and let cool.
When cool enough to handle, shred the chicken into bite‑sized pieces and season lightly with 1 teaspoon salt. Divide into 8 equal portions.
Add the remaining 1 tablespoon salt, the ginger, and the rice to the poaching liquid and bring to a boil. Stir, reduce the heat to a low simmer, and tightly cover with a lid.
Cook the rice for 2 hours or longer, stirring every 15 minutes or so, until it is very soft and thick like oatmeal.
Thinly slice the scallions.
Spoon the congee into 8 bowls and top each with a little pile of the shredded chicken and scallions.
Serve with baby bok choy or steamed broccoli florets for dipping.