Heat sugar and 3 Tbsp. water in a large saucepan over medium. Cook, stirring often, until sugar dissolves, about 2 minutes. Continue to cook, swirling pan occasionally, until syrup is golden amber, 4–6 minutes.
Add ginger and cook, stirring occasionally, until ginger is softened slightly, about 5 minutes. Add garlic and pale-green and white scallion parts and cook, stirring often, until garlic is fragrant, about 2 minutes. Stir in cilantro bundle, chicken, chile, and a big pinch of salt. Pour in 8 cups cold water and bring to a boil. Reduce heat to medium-low, bring to a bare simmer, and cook, stirring occasionally, until chicken is cooked through, 10–12 minutes. Discard chile and scallions. Keep soup warm over medium-low heat.