Preheat the broiler. Place tomatilos and serranos on a baking sheet. Broil turning once, until slightly charred, 5 to 7 minutes; discard chile stems.
In a blender, combine tomatillos, serranos, onion, cilantro, limejuice, and salt to taste and puree until well blended; set the salsa aside.
Heat large skillet over medium heat. Toast tortillas one at a time until puffed and very lightly browned, about 1 minute per side.
Lay tortillas on a work surface. On half of each tortilla, place one-fourth of the chicken, one-fourth of the cheese, and about 2 tablespoon tomatillo salsa. Fold tortillas in half.
In a lightly oiled skillet or grill pan over medium heat, cook one quesadilla at a time until crisp and golden brown on both Side dish. Add more oil if necessary for remaining quesadillas. Transfer to a cutting board and let cool slightly. Slice into wedges and serve with remaining salsa and sour cream, if desired.