In a bowl, mix together the avocado, cucumbers, sambal oelek, lime juice, salt and pepper.
Sandwich some of the chicken and cheese between two tortillas; repeat with the remaining chicken, cheese and tortillas. Brush the top and bottom of each quesadilla lightly with oil.
Heat a large, nonstick skillet over high heat until very hot. Cook the quesadillas, in batches, if necessary, until the cheese melts and the tortillas are crisp and golden, 1½ to 2 minutes per side.
Cut into wedges and top with some of the salsa.