Slice the zucchini and yellow squash into 1/2-inch thick rounds, keeping the portobello mushroom caps whole with stems removed, and cut the red bell peppers into quarters removing the seeds. Cut the red onion into 1/2-inch thick wedges. Pat all vegetables dry with paper towels, then season generously with kosher salt and freshly cracked black pepper on both sides. This prep work ensures even seasoning and helps the vegetables develop a nice char on the grill.
Heat your grill to 300-350°F. Place all seasoned vegetables directly on the grill grates. Grill for 3-4 minutes per side until you see nice char marks and the vegetables are tender. The peppers will take slightly longer and should be charred on the outside; this charring adds incredible depth of flavor. I like to grill the mushrooms last since they absorb heat differently than the other vegetables.
Transfer all grilled vegetables to a cutting board to cool slightly. Once the peppers are cool enough to handle, gently peel away the charred skin, then remove any remaining seeds and rinse under cool water. Chop the peppers into bite-sized pieces. Chop or slice the remaining vegetables (zucchini, squash, mushrooms, and onions) into manageable pieces for the quesadillas. Set all vegetables aside.
Lay out one flour tortilla on a clean surface and spread 2 tablespoons of basil pesto evenly over the entire tortilla. On one half of the tortilla, arrange about 1/4 of the chopped grilled vegetables, then sprinkle 1/4 of the mozzarella (about 3-4 tablespoons) and 2 tablespoons of parmesan cheese over the vegetables. Fold the tortilla in half to create a half-moon shape, pressing gently to secure. Repeat this process with the remaining three tortillas and ingredients.
Heat a large skillet or griddle over medium-high heat. Once hot, place one assembled quesadilla in the pan and cook for 3 minutes until the bottom is golden brown and crispy. Carefully flip and cook the other side for another 3 minutes until both sides are evenly browned and the cheese has melted. I recommend using a wide spatula to flip these—it makes the process much easier and prevents the filling from spilling out. Repeat with the remaining quesadillas, working in batches if needed to avoid overcrowding the pan.
Transfer the cooked quesadillas to a cutting board and let them rest for 1-2 minutes. Cut each quesadilla into four triangular pieces. If desired, drizzle lightly with balsamic glaze for a finishing touch. Serve immediately while the cheese is still warm and gooey.