In same skillet, heat 1 teaspoon oil over medium-high heat. Add 6 of the patties; cook 4 to 6 minutes, turning once, until thermometer inserted in center of patties reads 165°F. Remove patties from skillet to platter; sprinkle each with 2 teaspoons cheese. Cover to keep warm. Repeat with remaining 1 teaspoon oil, 6 patties and cheese.