Stir together caramelized onion jam, thyme, ¼ tsp salt, and ½ tsp pepper. Place tortillas on clean work surface and spread onion filling on half of each tortilla. Top with some apple and mozzarella, fold quesadillas in half, and gently press to seal.

Heat 1 tsp olive oil in large nonstick skillet over medium heat. Add one quesadilla and cook until golden brown, 2 to 3 minutes per side. Repeat with remaining quesadilla and oil. Cut into quarters and divide between plates.