Prepare the barbecue for medium-high heat.
Using a large heavy knife, mash the garlic with a pinch of salt into a paste. Transfer the garlic to a small bowl and add 1/2 cup of olive oil in a slow stream, whisking until combined well.
Brush the bread slices with 1/4 cup of the garlic oil. Grill until golden, about 5 minutes per side. Cut the bread into cubes.
Whisk the vinegar into the remaining garlic oil. Season the vinaigrette to taste with salt and pepper.
In a large bowl, toss the grilled bread, tomatoes, mozzarella, basil, arugula, and shallots with enough vinaigrette to coat. Let stand 10 to 15 minutes at room temperature to allow the flavors to develop, then toss again just before serving.