Preheat the grill over high heat. Allow the pizza dough to sit at room temperature for about 20 minutes.
Meanwhile, combine the cherry tomatoes, olive oil, 2 sprigs of basil, oregano, a generous pinch of salt, and a few cracks of black pepper in a 12” cast-iron skillet. Toss well to combine. Place the skillet onto the grill. Allow the tomatoes to cook for about 8-10 minutes, or until blistered and softened. Remove the skillet and let the tomatoes cool.
Drizzle the sliced eggplant with a generous coating of olive oil and season well with salt and black pepper. Grill the eggplant over high heat for about 2-3 minute per side, until charred and tender. Transfer the eggplant to a cutting board and cut into quarters.
Place a pizza stone (or an upside-down sheet pan) onto the grill. Cover the grill and allow the stone/pan to become piping hot. Roll each dough ball out into a 10” circle.
Working with one pizza at a time, lightly flour a pizza peel and place the dough onto the peel. Build the pizza by drizzling and brushing enough olive oil to coat the top of the dough. Distribute a ¼ of the fontina, mozzarella, cherry tomatoes, and grilled eggplant over top. Gently slide the pizza onto the hot pizza stone using the pizza peel. Cover the grill, and cook for approximately 8-10 minutes, or until the pizza dough is golden/crisp and the cheese is melted.
Keep the pizza warm as you repeat with the remaining dough. Toss the arugula with a drizzle of olive oil. Top each pizza with a handful of arugula, the Prosciutto di Parma, the remaining fresh basil leaves, and a pinch of flaky sea salt. Slice each pizza into 6 slices.