In a wok or frying pan, boil the chicken broth with one tablespoon sesame seed oil.
With the broth boiling briskly, drop in the chicken pieces. Stir once to separate. Cook just until the chicken meat changes color. Scoop out and transfer to a plate.
Bring the broth to a hard boil once more. Drop in the broccoli florets and cook for one minute. Scoop out and dump in a bowl of iced water. Leave for a minute and drain well.
Make the sauce by stirring together the 1/3 cup of the chicken broth, starch and sugar.
Clean out the wok or frying pan. Pour in the remaining sesame seed oil and heat. Saute the garlic and ginger with a bit of salt and pepper until fragrant and lightly browned.
Turn up the heat. Add the chicken and broccoli to the pan. Sprinkle with salt and pepper (add sparingly if the chicken broth is already seasoned). Stir fry for a minute.
Pour in the sauce. Cook, stirring, until thick and clear. Taste and add more salt and pepper, as needed.
Serve the chicken broccoli stir fry immediately.