Heat the oil in a wok over high heat until it just begins to smoke. Stir-fry the chicken chunks in the oil until they begin to brown. Reduce the heat to medium and add the soy sauce, vermouth, garlic and ginger, and stir-fry until the majority of the liquid in the wok is gone. Add the scallions and sesame seeds and sauté until the chicken is cooked and the sesame seeds are toasted. Arrange the minced watermelon over 4 to 6 plates and serve the chicken over the watermelon.