Bring a pot of water with sliced ginger, smashed garlic, and a pinch of salt to a simmer. Add chicken and cook 12 to 15 minutes until cooked through. Cool, then shred.

Toast the slivered almonds in a dry skillet over medium heat, shaking often, until golden and fragrant, 3 to 5 minutes. Transfer to a plate.

Whisk the soy sauce, rice vinegar, sesame oil, honey, grated ginger, and garlic until smooth. Taste and add salt if needed.

Combine the Napa cabbage, red cabbage, carrots, bell pepper, scallions, and shredded chicken in a large bowl. Pour half the dressing over and toss until lightly coated.

Transfer to a platter, drizzle the remaining dressing, scatter with toasted almonds and sesame seeds. Serve right away.