Mix mayonnaise, jalapenos, cumin powder, paprika and cayenne powder in a small bowl and set aside.
Lightly season chicken with salt and pepper. Add cornstarch and mix well by hand.
Heat oil in a nonstick skillet until warm. Spread chicken slices in the skillet and saute over medium heat. When the bottom side turns golden, in 1 to 2 minutes, flip the chicken slices. Cover and cook over low heat until the bottom side turns golden and chicken is cooked through. Transfer chicken to a plate and set aside.
To assemble quesadillas, place cheddar cheese on half a side of each tortilla. Add 2 to 3 pieces of chicken and scatter mozzarella cheese on top. Spread mayonnaise mixture on the other half side of each tortilla. Fold into a half moon shape. Assemble the rest of quesadillas in the same way.
Heat a clean nonstick skillet over medium heat. Grill quesadilla over medium heat, one side at a time, until both sides turn golden brown and the cheese is well melted. Transfer to a cutting board and cut into 3 slices.