Place vegetables on a large platter, drizzle with two tablespoons of olive and season with salt and pepper.
Heat a grill or a grill pan to medium-high (if you’re using an outdoor grill lightly oil the grill grates).
Grill vegetables, turning frequently, until tender and slightly charred, about 4 minutes for the onion and 7 minutes for the eggplant, zucchini, and red bell pepper.
Transfer vegetables to a cutting board and let cool for about 5 minutes. Cut into into roughly 1½-inch pieces and set aside.
In a large skillet sizzle ½ teaspoon of butter over medium-low heat.
Lay a tortilla, then build the quesadillas by spreading 1 tablespoon of pesto, then by laying some grated cheese on the tortilla, and then arranging some of the grilled veggies. Sprinkle a little more grated cheese and top with a second tortilla.
When the tortilla is golden, carefully flip the quesadilla to the other side, adding another ½ teaspoon butter to the skillet at the same time. Continue cooking until the second side is golden.