1 1- to 1 1/4-pound skirt steak
2 tablespoons fresh lime juice
4 tablespoons (about) olive oil
12 5- to 6-inch-diameter corn tortillas
2 cups (packed) grated hot pepper Monterey Jack cheese
1 1/2 cups Spicy Pico de Gallo or purchased salsa
Place steak in glass dish. Sprinkle steak on both sides with lime juice, salt and pepper. Let stand at least 15 minutes and up to 1 hour, turning steak occasionally.
Preheat oven to 375°F. Heat 1 tablespoon oil in heavy large skillet over high heat. Add steak and sauté to desired doneness, about 3 minutes per side for rare. Transfer steak to cutting board. Let rest 5 minutes; slice thinly.
Brush 4 tortillas with oil; place tortillas, oil side down, on 2 baking sheets. Spread 1/4 cup cheese and 1 tablespoon pico de gallo on each tortilla. Top each with second tortilla. Spread each with 1/4 cup cheese, 1/4 of steak and 1 tablespoon pico de gallo. Press third tortilla onto each stack. Brush top tortillas with oil.
Bake quesadillas 10 minutes. Using metal spatula, turn each over. Continue to bake until heated through and golden, about 10 minutes longer. Transfer quesadillas to plates. Cut into wedges. Serve with remaining pico de gallo.
Spicy Pico de Gallo
1pound plum tomatoes, seeded, chopped
1 cup chopped onion
6 tablespoons (packed) chopped fresh cilantro
1/4 cup fresh lime juice
2 large garlic cloves, minced
1 1/4 tablespoons minced seeded jalapeño chilies
Combine all ingredients in medium bowl; toss to blend well. Season to taste with salt and pepper. Let stand at least 30 minutes for flavors to develop.