Puree 3/4 cup basil leaves and 1/3 cup plus 2 tablespoons oil in a blender until smooth and only tiny flecks of basil remain. Set a fine-mesh strainer over a large bowl. Strain mixture into the bowl, pressing on solids to extract as much oil as possible. Discard solids in strainer.
Add shallot through lemon juice to the strained oil, whisking to blend, Season to taste. Dressing can be made one day ahead; bring to room temperature and re-whisk before using.
Slice tomatoes into assorted wedges, rounds, and cubes; add to bowl with dressing. Toss to coat; let marinate at room temperature for 30 minutes.
Meanwhile, build a medium-hot fire in a charcoal grill, or preheat a gas grill on high. Brush bread with remaining 1/3 cup olive oil, season with salt and pepper, and grill bread until slightly charred in spots, about 2 minutes per side. Rub grilled bread with cut sides of garlic clove, then tear bread into one- to two-inch pieces.
Add bread and remaining 3/4 cup basil leaves to bowl with tomato mixture; toss to coat. Adjust seasonings before serving.