Mix mashed potatoes, flour, eggs, 1 teaspoon sesame oil, salt, and white pepper together in a medium mixing bowl until doughy enough to form a ball in your hand.

Heat 1 tablespoon vegetable oil in a large frying pan over medium heat. Sauté scallions in hot oil until soft, about 5 minutes. Fold scallions into the potato mixture until combined.

Heat enough vegetable oil in the frying pan to cover by 1/4 inch and add a drizzle of remaining sesame oil.

Form balls of potato mixture using a 1/4 cup measure and flatten into pancakes. Gently place into hot oil in batches and cook until browned on both sides and firm, approximately 5 minutes per batch.

Transfer pancakes to a paper towel-lined plate to soak up excess oil. Repeat for remaining batches, adding more vegetable oil and sesame oil as needed.