Heat a large heavy skillet, preferably cast iron, over medium-high heat. Add the oil, swirl to coat the pan, and immediately add the sliced onion. Cook the slices, stirring frequently, until well browned (almost a little charred) at the edges but still crisp-tender, about 5 minutes. Adjust the heat higher, if necessary, so the onions cook quickly. Remove to a plate and wipe out the skillet. Turn the heat to medium.
To assemble the quesadillas, divide the onion among the tortillas, spreading them over half of each tortilla. Divide the shrimp among the tortillas, arranging it on top of the onion. Scatter the tomatoes and cilantro evenly over the shrimp. Arrange the cheese over the filling. Fold the tortillas over to form half-circles.
Heat the skillet over medium heat. Place 1 or 2 tortillas in the pan (depending on the size of the pan) and cook until the bottom is nicely browned, about 2 minutes. Use a wide spatula or 2 spatulas--sandwiching the tortilla between them--to turn the quesadillas. Brown the other side, making sure the cheese melts. Place on a cutting board. Repeat to make the other quesadillas.
Use a sharp knife to cut each quesadilla into 3 wedges. Arrange on warm dinner plates and serve immediately with salsa.