Make the dressing: Whisk together garlic, ginger, vegetable oil, Dijon mustard, toasted sesame oil, rice vinegar, lemon juice, soy sauce, salt, sugar, and pepper in a small bowl until combined. Chill if making ahead.
Marinate the chicken: Pour ½ cup of the dressing over chicken breasts, sprinkle with salt, and let it sit for at least 30 minutes at room temperature.
Toast almonds: Preheat oven to 350°F, toast almonds on a baking sheet for 8–10 minutes until golden. Chop and set aside.
Prepare chicken for baking: Remove chicken from marinade and arrange on a foil-lined baking sheet. Seal in a foil packet.
Bake the chicken: Bake for 30–40 minutes until the internal temperature reaches 150°F. Let cool, then shred into large pieces.
Prepare vegetables: Thinly slice purple cabbage, bok choy, and scallions. Drain and quarter water chestnuts, and set aside mung bean sprouts.
Assemble the salad: In a large bowl, toss shredded chicken, vegetables, and remaining dressing. Adjust seasoning with more salt if needed.
Serve: Transfer salad to a platter and top with toasted almonds. Enjoy!