Mix mashed potatoes, flour, eggs, 1 teaspoon sesame oil, salt, and white pepper together in a medium mixing bowl. The mixture should be a little doughy, enough to form a ball in your palm.
Heat 1 tablespoon vegetable oil in a large frying pan over medium heat. Saute scallions in the hot oil until soft, about 5 minutes. Fold 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 the hot oil in batches and cook until browned on both sides and somewhat firm, about 5 minutes. Transfer pancakes to a paper towel-lined plate to soak up excess oil. Repeat with remaining mixture, adding more vegetable oil and sesame oil before frying each batch. Transfer to a platter and serve hot.