Combine the chicken breast, the chicken wings, 1 cup dark soy sauce, the sesame oil, and the garlic in a mixing bowl; season with white pepper. Slice the Chinese sausages on an angle and add to the chicken mixture. Mix well, cover, and chill at least 10 minutes.

Thoroughly rinse the shiitake mushrooms and place in a bowl; pour hot water over the mushrooms; allow the mushrooms to sit submerged until they are soaked and bloated, about 15 minutes. Remove the mushrooms, reserving the liquid. Cut the stalks from the mushrooms and discard. Slice the mushrooms in half and set aside.

Heat the oil in a large, deep skillet over medium heat. Cook the chicken pieces in the hot oil until no longer pink in the center and the juices run clear, 7 to 10 minutes. Drizzle 1/2 cup dark soy sauce over the chicken.

Wash the rice with water until the water comes out nearly clear; drain completely. Combine the rice and 1 1/2 cup of the reserved liquid from the mushrooms in a non-stick pan; bring to a boil. Reduce heat to low; cover and allow to simmer covered for 10 minutes with the heat on. Add the chicken mixture and mushrooms and remove from the heat; allow to sit covered until the rice is completely tender, 15 to 20 minutes.

Stir together the chile paste, ginger, and lime juice in a small bowl; drizzle over the chicken rice and top with shredded lettuce to serve.