In a medium pot, combine the rice, chicken stock, ½ cup of water, 1 piece of ginger, and a big pinch of salt. Bring to a boil, then reduce the heat to low. Cook, stirring occasionally to keep the rice from sticking, 45 to 50 minutes, or until thickened. Turn off the heat. Carefully remove and discard the ginger.
While the congee cooks, in a bowl, combine the cucumbers, sesame oil, crushed red pepper flakes, and vinegar. Season with salt and pepper. Set aside to marinate, stirring occasionally.
Bring a pot of water to boil. Poach the chicken breasts until cooked through. Shred or slice the chicken.
Serve the congee topped with the chicken, marinated cucumbers, and scallions. Add soy sauce to taste.