1. In a large nonstick skillet, heat 1 tablespoon oil over medium-high. Add onion and garlic; season with salt and pepper. Cook, stirring occasionally, until soft, 3 to 5 minutes. Add mushrooms, and cook until mushrooms are browned, about 7 minutes. Transfer to a bowl and toss with lemon juice. Wipe skillet clean.

2. Divide prosciutto, Gruyere, mushroom mixture, and arugula among bottom halves of lavash bread or halves of tortillas; fold other half over filling to close. Heat remaining oil in skillet over medium-high and cook a quesadilla until cheese melts and lavash or tortilla is crisp, 1 to 2 minutes per side. Repeat with remaining quesadillas.