Bring water to a boil in a covered medium pot. Add tomatillos, onion and serrano; boil, covered, for 10 minutes.
Drain the vegetables and transfer to a blender. Add cilantro, broth, garlic, and 1/4 teaspoon salt; pulse until chunky, about 5 pulses (use caution when blending hot ingredients).
Pat chicken dry. Sprinkle with cumin, oregano and the remaining 1/8 teaspoon salt. Return 1 cup salsa verde to the medium pot and bring to a light boil over medium-low heat (reserve the remaining salsa verde for serving). Add the chicken, stirring to coat. Cook, covered and stirring once, until the chicken is cooked through, about 8 minutes.
Remove from heat and, using two forks, pull the chicken apart until completely shredded. Return to medium-low heat and continue cooking, stirring occasionally, until the sauce has thickened, 2 to 3 minutes.
To assemble quesadillas, spoon 5 tablespoons pulled chicken onto one half of a tortilla. Top with 3 tablespoons cheese and 1 tablespoon chiles; fold to close. Repeat with the remaining tortillas.
Heat a large nonstick pan over medium-low heat. Cook 2 or 3 quesadillas at a time, flipping once, until crispy and golden and the cheese has melted, about 6 minutes. Repeat with the remaining quesadillas. Serve with the remaining salsa verde.