Working in batches if necessary, place bread into a large skillet and drizzle with 1/4 cup olive oil. Cook and stir bread over medium heat, adding more oil as needed, until golden and crispy on the outside, 7 to 12 minutes.

Sprinkle Parmigiano-Reggiano over bread cubes; cook and stir until cheese is melted and bread is crispy on the outside but still chewy on the inside, 3 to 4 minutes more. Let cool completely, 10 to 15 minutes.

Mix tomatoes, vinegar, 2 tablespoons extra-virgin olive oil, garlic, sugar, salt, and pepper together in a bowl until well combined. Cover the bowl with plastic wrap and let sit at room temperature until flavors combine, about 20 minutes.

Toss basil into tomato mixture, then stir in bread cubes. Let sit at room temperature for 3 to 4 minutes; if the liquid is completely absorbed, add more vinegar and olive oil to taste. Season with salt and black pepper.