Place a small saucepan over medium-high heat until hot. Add 3 tablespoons oil, swirling to coat sides. Add ginger, garlic, and green onions; cook, stirring, for 1 minute. Add vinegar, soy sauce, honey, chili garlic sauce, and sesame oil. Cook, stirring, until mixture comes to a boil. Stir in pickled ginger. Remove from heat and set aside.