Crack garlic and trim root, then halve pieces for the chicken and reserve a few for broccolini. Peel the 3 inches ginger and slice 2 inches into matchsticks or thin julienne for chicken and leave a small chunk to grate for broccolini.
Gather ingredients for broccolini.
For the broccolini, trim the ends and chop broccolini. Chop the leek into 1/4 inch dice. Slice garlic, grate ginger and chop the chili peppers. Heat the oil for the broccolini in a large skillet over medium-high to high heat. Add the broccolini and cook 1-2 minutes, add leeks, garlic, ginger, chopped chili peppers and sesame oil, toss 2-3 minutes more, add sauce and water, then toss, cover, remove from heat. Sprinkle with sesame seed to serve.
To cook chicken, heat cast-iron skillet over medium-high heat with oil, 3 turns of the pan. Cook the chicken pieces to brown 4-5 minutes, turn and add halved garlic and matchsticks or thin julienne of ginger. Brown 2-3 minutes and remove the chicken to a plate, add rice wine or sherry, stir in jam to melt, add stock, hot sauce and soy sauce, thicken a bit, then add the chicken back to pan and glaze. Turn off heat and cover.
Meanwhile, heat water and salt and rice to a boil, cover and cook about 15 minutes, then turn off heat and fluff with fork.
Serve Dragon Chicken, Broccolini with Black Bean Sauce and Sesame and Jasmine Rice family-style.