Place 8 tortillas on a work surface. Divide the cheeses, lobster meat, red onion, garlic, jalapeños, and cilantro among the tortillas. Stack the tortillas two high, to make 4 two-layer quesadillas.
Remove them, break off the legs and claws, and shock the lobster bodies in ice-cold water. Return the legs and claws to the boiling water for a further 2 minutes, then shock in ice-cold water.