Combine the chicken and ginger in a medium saucepan and cover with water by 1 inch. Bring to a boil over medium-high heat, then reduce the heat to medium and simmer until the chicken is cooked through, about 10 minutes. Drain and transfer the chicken to a cutting board.
Meanwhile, make the dressing: Whisk the lime juice, peanut oil, brown sugar, fish sauce and Sriracha in a small bowl; set aside.
Combine the cabbage, carrots, snow peas and mint in a large bowl. Add all but 2 tablespoons of the dressing and toss to coat.
Thinly slice the chicken. Top each serving of salad with the chicken and peanuts; drizzle with the remaining dressing.