Using a mallet or rolling pin, pound chicken breasts between two pieces of waxed paper to 3/4-inch thickness. Brush chicken with a little olive oil and season with salt and pepper.
Grill chicken, turning once, until an instant-read thermometer inserted into the thickest part registers 165 degrees, about 4 to 5 minutes per side. Transfer chicken to a plate, and when cool, shred or slice into super-thin slices. Set aside. ?
Meanwhile, combine sauce ingredients in a medium saucepan. Bring to a simmer and cook for 10 minutes. Combine sauce and chicken; stir to coat.
Combine mayonnaise, vinegar and sugar in a medium bowl. Add cabbage, carrots and onions; toss to combine.
Place about 1/2 cup chicken on top of a tortilla on a plate. Sprinkle with 1/4 cup cheese. Place another tortilla on top. Using a wide spatula, slide assembled quesadilla onto hot grill and cook until bottom is brown and crispy, about 2 minutes. Carefully flip and continue cooking until underside of the second tortilla is also brown and crispy, about another 2 minutes.
Place cooked quesadilla onto a cutting board and cut into wedges. Repeat with the remaining ingredients to make three more quesadillas. Serve with coleslaw.