Peel the oranges with a knife. Segment the oranges and transfer the segments to a small bowl. Set aside for garnishing the salad. Squeeze the juice from the orange membranes into a second bowl (juice should measure ¼ cup).
Combine the orange juice, vinegar, soy sauce, ginger, sugar, and chili-garlic sauce in a bowl. Transfer ½ cup of the orange juice mixture to a 12-inch skillet. Slowly whisk vegetable oil and sesame oil into the remaining orange juice mixture to make the vinaigrette. Set aside.
Bring the orange juice mixture in the skillet to a boil. Add the chicken, reduce heat to medium-low, cover and simmer until the meat registers 160 degrees (about 10-15 minutes), flipping halfway through cooking. Transfer the chicken to a plate and let rest for 5-10 minutes.
Meanwhile, boil the pan juices until reduced to ¼ cup. This will only take about 1-3 minutes. Set aside. Shred the chicken into bite size pieces, transfer to a medium bowl, and toss with the reduced sauce, any accumulated juices, and 2 tablespoons of the vinaigrette. Let sit for 10 minutes.
Toss the romaine, cabbage, bell peppers, cilantro, peanuts, and scallions with the remaining vinaigrette in a large bowl. Transfer to a serving platter and top with the chicken and oranges.