First of all, forget everything you think you know about quesadillas. A microwaved flour tortilla stuffed with cheddar cheese is about as authentically Mexican as a Dorito Loco taco from Taco Bell. When my wife and I lived in Mexico, we discovered that the soul of real Mexican food is masa, the ground-corn dough — not wheat flour — that's used to make soft corn tortillas for tacos, steamed tamales, gorditas, and innumerable regional antojitos, Mexican street snacks with deep indigenous roots.
A real Mexican quesadilla starts with that freshly ground corn masa. A large hunk of soft dough is pressed or rolled into a rough circle and laid on a hot griddle. The basic version is filled with shredded Chihuahua or Oaxacan string cheese and folded over into a half moon. But why stop with cheese? At Mexican street stalls, you'll find mouth-watering fillings like potato and crumbled chorizo sausage, nopal (cactus), squash blossoms, sautéed mushrooms, huitlacoche(an earthy corn fungus), refried beans, and all varieties of grilled or slow-cooked meats.
I've found that real-deal quesadillas are best enjoyed on a wobbly wooden bench inside a bustling open-air market with a healthy dollop of salsa verde (the tomatillo-based green salsa) and an icy agua fresca, a sugary fruit juice in every imaginable flavor ladled from giant plastic jugs.