Cook onion in olive oil in heavy skillet over medium heat, stirring, until softened, about 5 minutes. Transfer onion to a bowl. Remove pan from heat and set aside.
Break up turkey, crumbling meat into bowl with onion. Sprinkle salt and pepper on top, then gently toss meat and onion together to blend. Form turkey mixture into twelve 2-inch patties, about 1/2 inch thick.
Arrange open rolls on a platter and top each bottom with a lettuce leaf.
Heat skillet used for onion over medium heat until hot. Add 6 burgers and cook for 3 minutes. Turn burgers over, place a slice of cheese on top, cover, and cook for about 3 minutes more, until burgers are cooked through. Put burgers on lettuce-lined roll bottoms, followed by roll tops. Repeat with remaining burgers.