Trim the knobby ends and sides off the ginger. Place these ginger trimmings in a large Dutch oven or another heavy pot with the chicken. Add 3 cups of cold water and a generous pinch of salt. Bring the mixture to a boil over high heat. Once boiling, reduce the heat to maintain a gentle simmer. Allow the chicken to simmer, turning the pieces for about 15 minutes. While the chicken is cooking, prepare the remaining ingredients:1. Rinse the rice in a sieve until the water runs clear, then drain it.2. Finely chop the trimmed ginger.3. Trim and finely chop the scallions.4. Combine the chopped ginger and scallions in a heatproof bowl, season generously with salt, and set aside. After the chicken has simmered for 15 minutes, taste the broth and adjust the seasoning with salt to make it savory. Stir the rinsed rice into the pot. Use tongs to arrange the chicken pieces skin side up over the rice. Increase the heat to high to bring the water to a boil, then cover the pot and reduce the heat to low. Allow it to cook until the rice is tender, about 20 minutes. Please turn off the heat and let it rest, covered, for 5 minutes before serving.
As the rice rests, heat the oil in a wok or large skillet over high heat until it shimmers, which should take 1 to 2 minutes. Gently pour the hot oil over the chopped ginger and scallions, ensuring a thin layer of oil remains in the pan.
Place the wok back on high heat and add the greens and a pinch of salt. Cook, stirring constantly, until the greens become vibrant in color and are just tender. If the greens appear dry, add a splash of water to the pan.
Mix the vinegar into the ginger-scallion sauce. Taste and adjust the seasoning with more salt if necessary. Serve the sauce alongside the chicken rice and stir-fried greens. Remember to remove the ginger trimmings from the rice while eating.