In a medium bowl combine egg, rolled oats, half of the oregano, the salt, and black pepper. Add ground turkey breast; mix well. Shape turkey mixture into four 3/4-inch-thick patties.
For a charcoal grill, grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until no longer pink (165 degrees F), turning once halfway through grilling and topping each burger with a cheese slice for the last minute of grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place burgers on grill rack over heat. Cover and grill as above.)
Meanwhile, in a small saucepan, cook pasta sauce until heated though.
Place burgers on the bottom halves of buns. Top with pasta sauce, the remaining half of the oregano, and the top halves of buns.