Chicken Quesadillas

Sprinkle chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper. Cook onion with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper in oil in a 10- to 12-inch skillet over moderate heat, stirring occasionally, until golden, about 6 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute, then transfer to a large bowl. Add chicken to onion mixture along with cheese. Put 1 tortilla on a cutting board and spread 1/2 cup chicken mixture over half of tortilla, then fold other half over to form a half-moon, pressing firmly on seam. Assemble 7 more quesadillas in same manner. Heat lightly oiled grill pan over high heat until it begins to smoke, then reduce heat to moderate and grill quesadillas, 2 at a time, turning over once, until cheese is melted and golden brown grill marks appear, about 4 minutes total per batch. Transfer with a spatula to cutting board and cut in half.