Roast Chicken Noodle Soup with Chrysanthemum

3 cups water

1 2-inch piece fresh ginger , peeled, sliced into thin rounds

6 cups low- salt chicken broth

2 tablespoons soy sauce

2 teaspoons sugar

3 tablespoons vegetable oil

3 garlic cloves, thinly sliced

2 bunches chrysanthemum, bottoms trimmed, upper stems and leaves cut into 2-inch strips (about 5 cups)

1 14-ounce package thin fresh or dried Chinese egg noodles

2 1/2 cups 1/2-inch cubes skinned and boned freshly roasted or purchased roast chicken

1/4 large onion , sliced paper-thin

3 red Thai bird chiles or 1 large red jalapeño chile, sliced into thin rounds