Heat 1 tbsp oil in large, heavy skillet over medium-high heat. Add onion and serrano chile; cook, stirring occasionally, until soft and lightly browned, about 5 minutes. Add garlic and stir-fry 1 minute. Sprinkle with a pinch of salt and pepper. Transfer to a medium bowl and wipe skillet clean with a paper towel.
Chope chipotle chile; place in a small bowl with adobo sauce and mash with fork. Add to onion mixture in bowl; add chicken and toss. Season to taste with salt and pepper.
Place a tortilla on a cutting board and spread with 1 cup chicken mixture. Top with 1/2 cup cheese and another tortilla. Repeat with remaining ingredients.
With paper towel, lightly oil skillet and place over medium-high heat until it is very hot but not smoking. Cook quesadillas one at a time, turning over once, until cheese is melted, 4-5 minutes total.
Transfer with spatula to cutting board; cover with aluminum to keep warm. Cut the quesadillas into quarters - serve with your choice of accompaniments (source cream, salsa, etc).