Preheat a grill to medium high. Make the salsa verde: Pulse 1 3/4 cups cilantro, the scallions, lime juice, vinegar, Worcestershire sauce and a large pinch each of salt and pepper in a food processor until slightly chunky. With the motor running, pour in the olive oil and puree until combined, scraping down the sides of the processor as needed. Season with more salt and pepper.
Poke the steak all over with a fork to tenderize it, then cut into 1-inch cubes; toss with 2 tablespoons of the salsa verde and season with salt and pepper. Thread onto eight metal skewers, finishing each skewer with a lime wedge; set aside.
Tear off a 16-inch piece of foil. Place 1 tortilla on the left side of the foil; top with the cheese and the remaining 1/4 cup cilantro, then sandwich with the other tortilla. Fold the right side of the foil over and seal the edges to form a packet.
Brush the grill grates with olive oil. Grill the kebabs, turning occasionally, 4 to 6 minutes for medium rare. Transfer to a plate and let rest. Meanwhile, grill the quesadilla until the cheese melts, about 2 minutes per side; unwrap and cut into wedges. Serve the kebabs in the lettuce leaves with the remaining salsa verde, the quesadilla wedges and mango slices.