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MAKE-AHEAD TIP: Garlic oil can be made up to 6 hours ahead, covered and refrigerated. Bring to room temperature before using.
- Heat grill to medium-high.
- Using a pestle and mortar or a large heavy knife, mash garlic with a pinch of salt into a paste. Transfer garlic to a small bowl and add 1/2 cup of olive oil in a slow stream, whisking until combined well.
- Brush bread slices with 1/4 cup of garlic oil. Grill until golden, about 5 mins per side. Cut bread into cubes.
- Whisk vinegar into remaining garlic oil. Season vinaigrette to taste with salt and pepper.
- In a large bowl, toss grilled bread, tomatoes, mozzarella, basil, arugula, and shallots with enough vinaigrette to coat. Let stand 10 to 15 mins at room temperature to allow flavors to develop, then toss again just before serving.