Bring 2 cups of very lightly salted water to a boil. When the water is boiling, add the green onions, half of the ginger, and the chicken. Add water as needed to cover your chicken. When the water comes back up to a boil, drop the heat to low and poach, keeping the liquid below a boil, for 25 minutes.
Meanwhile, make your dressing. Combine the garlic, chili, sugar, lime juice, and fish sauce in a jar, along with the remaining ginger and 1 cup of cold water. Stir or shake well to mix, then set aside.
When the chicken is done, remove the thighs to a bowl or cutting board. Save the chicken broth for another use. Shred the chicken.
In a large platter or salad bowl, combine your cabbage, onion, mint, cilantro, and shredded chicken. Top with the dressing, and then toss well. Top with your cubed avocado(es) before serving.