Toss the coleslaw mix with 1/2 teaspoon salt in a large bowl and set aside. Whisk the lime juice, garlic, cilantro, vegetable oil, 1/4 teaspoon salt and a few grinds of pepper in a small bowl.
Lay out the tortillas and sprinkle the cheese on half of each. Season the chicken with salt and pepper and layer on top of the cheese. Fold the tortillas in half. Heat 1 tablespoon butter in a large nonstick skillet over medium heat. Add 2 quesadillas and cook, turning once, until browned and crisp and the cheese is melted, 5 to 7 minutes. Transfer to a cutting board and repeat with the remaining butter and quesadillas.
Cut the quesadillas into wedges and divide among plates. Add the lettuce and lime dressing to the coleslaw mix, toss to combine and season with salt and pepper. Add to the plates and drizzle the quesadillas with the crema.