If you haven't already, separate all the "good meat" from the bones and set aside/refrigerate.
Place the picked over rotisserie chicken bones, skin and any accumulated and solidified juices into a large pot with a lid. Fill the pot with 4 quarts cool clean water or more to cover the bones by at least 1-inch. Set the pot over medium-high heat to begin the cooking process while you prepare the aromatics.
While the chicken in the pot is coming up to a boil, turn on another one of the burners on your gas stove to medium-high. Using a pair of metal tongs (and wearing a heatproof mitt or glove), place the onions and ginger directly on the burner over the flame, turning them occasionally until they are charred black all over, about 10 minutes. If you don't have a gas stove, you can get the same effect with a grill, the broiler in the oven, possibly a toaster oven. Place the charred onion and ginger along with the remaining soup base ingredients, garlic cloves, salt, fish sauce, sugar, cilantro stems, and spices, into the pot with the chicken, which should be about to boil, if not already. Bring to a boil, reduce heat, and simmer for 45 minutes to 1 hour.
Strain the soup base through a colander or a large sieve, pressing down on the skin, bones, onions, ginger and spices to extract all the flavorful liquid. Discard the skin, bones, onions, ginger, and spices. Taste the soup base and adjust the seasoning with any more of salt, fish sauce, or if you must, sugar.