Spread about 1 1/2 Tbsp. pesto on each tortilla. Sprinkle a slightly heaping 1/3 cup chicken onto half of each tortilla; sprinkle cheese over chicken on each tortilla.
Fold each tortilla in half. Butter both sides of each folded tortilla.
Heat a large nonstick skillet over medium-high heat. Cook quesadillas, in 2 batches, 2 minutes on each side or until browned and crusty. Remove to a cutting board, and cut each quesadilla into 3 wedges.