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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.
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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.
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Mix tomatoes, vinegar, 2 tabelspoons 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.
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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.