Fit the food processor with the chopping blade and put the lid in place. Quarter and core the tomatoes; using your fingers, remove the seeds. Halve the lime; trim, quarter and peel the onion; then trim the garlic clove and smash it.
Juice the lime with a citrus juicer, working over a small bowl (A). In the food processor with the machine on, add the garlic through the feed tube (it will stick to the side of the bowl). Add the onion through the tube, processing until chopped. Turn off the machine and remove the lid. Add the tomato pieces and the lime juice; attach the lid and pulse with 1-second pulses until chopped but not soupy. Give your salsa a taste test, just like a real chef! Taste a spoonful (B), then add just a little seasoning: 1/2 tsp. salt and 1/4 tsp. pepper. Pulse for 1 second to blend. Taste again! Add more salt if you think it needs it. Transfer the salsa to a bowl.
Discard the chicken skin. Using your fingers, shred the meat (C). Shred the cheese with a box grater.
Preheat a large cast-iron skillet over medium-low heat. Meanwhile, place a tortilla on a work surface and spread with 1 tbsp. cream cheese. Sprinkle 1/3 cup shredded cheese on top. Arrange 1/3 cup chicken on half of the tortilla (D) and fold in half. Repeat the process with the remaining tortillas. Drizzle a little EVOO in the hot skillet, add 1 or 2 quesadillas to the pan and cook, turning once or twice, until golden-brown and crisp on both sides, about 5 minutes. Repeat with the remaining tortillas. Cut the quesadillas in half, stack on plates, and serve warm with the salsa and yogurt.