Trim gizzards and cut in half. In a saucepan, combine gizzards with stock, whole scallion, and ginger slice. Bring to a boil, cover, reduce heat to medium low, and simmer until gizzards are tender, about 40 minutes. Remove from heat. In a wok, heat oil until just smoking. Add minced scallion and minced ginger. Stir fry over high heat until aromatic, about 30 seconds. Add oyster sauce, sugar, gizzards, and any stock remaining in pan. Bring to a boil, reduce heat, and simmer until gizzards have absorbed flavor and sauce has reduced slightly, about 15 minutes. Drizzle with sesame oil and serve.