Heat half the oil in a large skillet over medium-high heat until it shimmers but does not smoke. Add the zucchini and salt, then cook until the zucchini is browned and crisp-tender, about 4 to 5 minutes.

Transfer the zucchini to a medium bowl and stir in the jalapeños. Let the mixture sit for a few minutes to cool slightly; this prevents the steam from making the tortillas soggy during assembly.

Arrange the tortillas on a clean work surface. Distribute the shredded cheese and the zucchini mixture over half of each tortilla, then fold the tortillas over to enclose the filling.

Wipe out the skillet with a paper towel and heat the remaining oil over medium heat. Place two quesadillas in the skillet and cook until the bottom is golden brown and crisp, about 2 minutes. Flip carefully and continue to cook until the second side is golden and the cheese is fully melted.

Repeat the cooking process with the remaining tortillas. Slice into wedges and serve immediately, while the cheese is still hot and stretchy.