Prepare a medium-high (425°F to 450°F) gas or charcoal grill fire. Brush the pepper quarters with oil and season with salt and pepper. Brush both cut sides of the ciabatta lightly with oil.
Grill the peppers skin-side down until grill marks form on one side, about 5 minutes. Flip the peppers and continue to grill until crisp-tender, about 2 minutes more. Meanwhile, grill the bread, turning once, until nicely toasted on both sides.
Cut the bread into 3/4-inch cubes and transfer to a medium bowl. Cut the peppers into 1-inch pieces and add to the bowl. Stir in the cherry tomatoes and basil.
In a small bowl, whisk together the remaining 1/4 cup oil, the lemon juice, garlic, 1/4 tsp. salt, and 1/4 tsp. pepper. Drizzle over the salad and toss to combine. Season to taste with more salt and pepper and let stand for 20 minutes before serving so the flavors meld.