Put a cactus paddle flat-side down on a work surface. Scrape a knife almost horizontally from narrow base of paddle toward wider end to cut off thorns. Scrape off thorns on each side of the paddle and trim all around edges. Repeat with remaining paddles.
Heat a griddle over high heat and cook the cactus paddles until grill marks appear, flip them over. Top eight cactus paddles with string cheese and top cheese with the other eight paddles. When cheese is melted, cut each quesadilla in triangles and serve on a platter.
SALSA: In a non-reactive bowl combine tomatoes, onion, chiles, cilantro, salt and lime juice. Stir well. Let rest in the fridge for one hour before serving.