Rinse and soak the rice: rinse the jasmine rice 2 to 3 times until the water runs mostly clear. Cover with fresh water and soak for 30 minutes if you have time. Drain well.
Start the pot: put the drained rice, chicken thighs, sliced ginger, and 8 cups of water or stock into a large heavy pot. Bring to a boil over medium-high heat and skim off any foam.
Lower to a lazy simmer: reduce the heat to low so the surface barely bubbles. Partially cover and cook for 35 to 40 minutes, stirring every 10 minutes and scraping the bottom to prevent sticking.
Shred the chicken: lift the chicken out, discard skin and bones, and shred the meat into bite-size strands. Return the shredded chicken to the pot.
Season and adjust: stir in salt, white pepper, and fish sauce or soy sauce. Simmer another 8 to 10 minutes. Add another 1/2 to 1 cup hot water or stock if the congee is thicker than you like, until it flows slowly off a spoon.
Serve with toppings: ladle into warm bowls. Top with julienned ginger, scallions, cilantro if using, a pinch of white pepper, and a few drops of sesame oil. Set out crispy shallots and chili oil at the table.