Cut thin slices from the top and bottom of the oranges, then slice off the rind and pith. Working over a bowl, cut orange segments from the thin membrane and transfer segments to a second bowl; set aside. Squeeze juice from the membrane into the first bowl (juice should measure ¼ cup).

Combine orange juice, vinegar, soy sauce, ginger, sugar, and chili-garlic sauce in a bowl; transfer ½ cup of orange juice mixture to a 12-inch skillet. Slowly whisk vegetable oil and sesame oil into the remaining orange juice mixture to make a vinaigrette; set aside.

Bring orange juice mixture in skillet to boil. Add chicken, reduce heat to medium-low, cover, and simmer until meat registers 160 degrees, 10 to 15 minutes, flipping halfway through cooking. Transfer chicken to a plate and let rest for 5 to 10 minutes.

Meanwhile, boil pan juices until reduced to ¼ cup, 1 to 3 minutes; set aside. Using 2 forks, shred chicken into bite-size pieces. Off heat, add chicken, any accumulated juices, and 2 tablespoons vinaigrette to skillet. Toss to coat and let sit for 10 minutes.

Toss romaine, cabbages, bell peppers, peanuts, and scallions with the remaining vinaigrette in a large bowl. Transfer to a serving platter and top with chicken and oranges. Serve.