Prepare the grill for direct cooking over medium heat (350° to 450°F). Preheat the griddle over direct medium heat for 5 minutes.
Prepare the grill for cooking over medium heat (400°F). Preheat the griddle over direct medium heat for 5 minutes.
Cut two (4-inch) sections of the ciabatta loaf. Set aside the remaining bread for another use. Halve each bread section horizontally. Brush the cut sides of the bread with the aioli. Arrange a layer of mozzarella, ham, and arugula on the two bottom pieces of bread. Cover with the bread top.
Place the sandwiches on the griddle and grill, with the lid closed, until the bread is toasted, and the cheese begins to melt, 2 to 3 minutes. Using a spatula, flip the sandwiches. Grill, with the lid closed, until the bottoms are toasted, 2 to 3 minutes more. Serve at once.