Poach the chicken: cover the chicken with cold water in a pot, bring to a gentle simmer (not a rolling boil), and cook until the internal temperature reaches 165°F (74°C) — about 15 minutes for breasts, depending on thickness. Rest 10 minutes, then shred by hand. Save the poaching liquid for soup if you like.
Shred the cabbage thin — about 1/8 inch / 3 mm ribbons. A mandoline or sharp knife both work. Thicker shreds make the salad chewy and hard to dress evenly.
Whisk the dressing ingredients until the sugar dissolves. Taste: it should be bright and savory, not just salty.
Soak the sliced red onion in cold water for 10 minutes, then drain — this takes the harsh edge off.
Dressing timing: toss the cabbage with about two-thirds of the dressing just before serving (within 10 minutes). Add the shredded chicken, drained onion, and herbs; toss gently.
Top with peanuts and fried shallots. Taste and add the remaining dressing if needed. Serve immediately.