Puree chiles in adobo sauce and water in a blender until smooth. Pour mixture over chicken to coat. Refrigerate for 1 hour.
Meanwhile, place onion in a small bowl. Combine jalapenos, vinegar, and sugar in a small saucepan over medium heat. Bring to a simmer, and cook until jalapenos are soft and sugar has dissolved, about 10 minutes. Pour over onion, and let stand until mixture is cool. Strain through a fine sieve.
Heat a grill pan over high heat. Grill chicken, about 2 minutes per side. Slice into strips. Wipe pan clean.
Place 1 tortilla onto a work surface. Arrange strips of 1 cutlet to cover tortilla. Sprinkle with 1/3 cup Monterey Jack cheese and some onion-jalapeno mixture. Top with another tortilla. Repeat with remaining tortillas, chicken, cheese, and onion mixture.
Carefully place quesadilla onto hot grill pan, and cook over medium heat until tortillas are crisp and cheese melts, about 4 minutes per side. Cut each quesadilla in half, and divide among 8 plates. Cut into wedges.