Preheat grill to medium-high heat. Brush sliced vegetables with oil and season generously with Kosher salt. Grill, flipping once, until vegetables are slightly charred and cooked through, about 6-10 minutes. Remove vegetables from grill and roughly chop.
Place mozzarella cheese on one half of each flatbread. Top with a mixture of grilled vegetables and a drizzle of pesto. Fold flatbread into a quesadilla shape and grill approximately one minute per side, flipping carefully. Remove from grill, cut into pieces and devour immediately with extra pesto on the side for dipping.
for the Basil Pesto
Place the garlic in a food processor and chop. Add in the basil and pine nuts and process until finely chopped. With the motor running, drizzle in the oil. Transfer the mixture to a small bowl and stir in Parmesan cheese. Season with Kosher salt and freshly ground pepper. If making ahead of time, top pesto with a thin layer of extra virgin olive oil to prevent browning and refrigerate. Bring to room temperature and stir before serving.