Preheat oven to 350°F.
Chop green onions, grate carrots, shred cabbage, and drain oranges. Set aside separately.
Combine 1 tablespoon soy sauce and 1/2 teaspoon sesame oil and brush onto chicken breasts.
Arrange chicken in a baking dish and bake until juices run clear, about 15 minutes.
Remove from oven, cool completely, and cut into 1/4-inch slices.
In a large bowl, combine cabbages, carrots, green onions, water chestnuts, oranges, and chicken.
In a separate bowl, whisk together the remaining soy sauce, vinegar, minced garlic, freshly grated ginger, olive oil, remaining sesame oil, brown sugar, and chili sauce.
Pour dressing over salad and toss to combine; top with almonds before serving.