Preheat oven to 350 degrees F.
Combine 1 tablespoon soy sauce and 1/2 teaspoon sesame oil and brush onto chicken breasts. Arrange in a baking dish and bake until juices run clear, about 13 to15 minutes. Remove from oven, cool completely, and cut into 1/4-inch slices.
In a large bowl, combine Napa cabbage, red cabbage, carrot, scallions, water chestnuts, Mandarin orange and sliced chicken. In a separate bowl, whisk together 3 tablespoons soy sauce, vinegar, garlic, ginger, oil, 1 1/2 teaspoons sesame oil, brown sugar and chili sauce. Pour dressing over salad and toss to combine. Divide among bowls and top each serving with 2 teaspoons toasted almonds.