Heat the oven to 400°F. Tear the bread into 1-inch pieces, pile them on a rimmed baking sheet, and drizzle with 2 Tbs. of the oil. Sprinkle with salt, toss, spread in a single layer, and bake, turning occasionally, until the bread cubes are golden and crunchy, 8 to 12 min. Set aside to cool.
Bring a saucepan of water to a boil. Add the basil leaves and lemon zest and blanch for 5 seconds. Drain. Transfer the basil and zest to a blender and pulse a few times to chop the leaves. With the blender running, slowly add the remaining 1/3 cup oil through the lid’s fill hole and process for 30 seconds. Scrape down the sides of the blender and continue to purée until smooth, 30 to 60 seconds.
In a large bowl, toss together the cherry tomatoes, mozzarella, and bread cubes. Let sit, tossing occasionally, for 20 min. Sprinkle with the chile flakes; season with salt and pepper to taste.