Slice off the top third of each roll. Slice remaining portions horizontally into three slices each. Preheat panini machine, or heat a large cast-iron skillet over medium-high heat.
Spread three of the slices of ciabatta with pesto. Cover pesto with three slices of mozzarella each. Spread pepperonata evenly over mozzarella. Drizzle with oil, and sprinkle with salt and pepper. Cover with remaining three slices ciabatta.
Place sandwiches into machine. Close machine, and cook until bread is lightly toasted and mozzarella is hot but not completely melted, about 5 minutes. Alternatively, place sandwiches in heated skillet, and weight with a heavy pan lid, then cook until golden brown on both sides, turning once after about 4 minutes.