Place the chicken broth, water, chicken thighs, and sliced ginger in a large pot or Dutch oven. Bring to a gentle boil over medium-high heat, then lower the heat to maintain a steady simmer.
Simmer until the chicken reaches an internal temperature of 165°F (74°C), about 8 to 10 minutes. Transfer the chicken to a clean plate or bowl. Keep the broth simmering.
Add the cooked rice to the pot. Stir well, breaking up any cold clumps with a spoon. Simmer uncovered, stirring often and pressing some of the rice grains against the side of the pot, until the mixture becomes thick, creamy, and porridge-like, about 18 to 20 minutes.
While the congee simmers, use two forks to shred the chicken into bite-size pieces. Return the shredded chicken to the pot along with the soy sauce, salt, white pepper, and half of the scallions.
Cook for 2 minutes, stirring, until the chicken is hot throughout. If the congee becomes thicker than you like, stir in a splash of hot water or broth until it reaches your preferred consistency.
Remove the pot from the heat and stir in the sesame oil. Taste and adjust the salt if needed. Divide among bowls and top with the remaining scallions, optional cilantro, julienned ginger, and chili oil or chili crisp.