Pulse the bread in a food processor into large crumbs, about the size of a pea. Combine the crumbs, oil, and a pinch of salt and pepper in a nonstick skillet over medium heat. Cook, stirring often, until the crumbs are crisp and golden brown, about 10 minutes.

Clear the center of the skillet and add the garlic. Cook, mashing it into the skillet, until fragrant, then stir the garlic into the crumbs. Transfer everything to a plate and let the crumbs cool completely to maintain their crunch.

Spread the tomatoes over a baking sheet lined with several layers of paper towels. Sprinkle with salt and let sit for 30 minutes to draw out excess moisture.

Combine the shallot, vinegar, and salt in a large bowl. Whisking constantly, slowly drizzle in the remaining oil until the dressing is emulsified.

Toss the tomatoes and basil gently into the dressing, then arrange them on a serving platter.

Place the burrata on a clean plate and cut into 1-inch pieces, being careful to collect any creamy liquid that escapes. Arrange the cheese over the tomatoes, drizzle with the reserved creamy liquid, and top with the garlic bread crumbs. Serve immediately, while the crumbs are still crackling.