Thinly slice mushrooms and finely chop capsicum. Place mushrooms and capsicum in a medium bowl with spinach. Mix until combined.
Spoon the spinach mixture onto 6 wraps. Sprinkle with cheese. Place the remaining 6 wraps on top and press down gently.
Lightly spray the top and bottom of the sandwich press with cooking oil spray. Using a plastic egg flip, carefully place 2 quesadillas on the sandwich press and close the lid. Cook for 2 minutes or until browned on both sides and cheese is melted. Transfer quesadillas to a plate and cover to keep warm. Repeat cooking remaining quesadillas, lightly spraying the sandwich press in between batches with cooking oil spray.
Cut quesadillas into wedges. Serve with lime halves.