In a large mixing bowl, combine the diced tomatoes, torn basil, minced garlic, 1/4 cup olive oil, and balsamic glaze. Season with 1/2 teaspoon kosher salt and several grinds of black pepper. Stir well and set aside at room temperature to let the flavors meld while you cook the pasta.

Bring a large pot of water to a rolling boil. Add 2 tablespoons kosher salt, then add the pasta. Cook according to package directions until al dente, usually 10-12 minutes. The pasta should still have a slight bite in the center. Reserve 1/2 cup pasta water, then drain immediately and rinse briefly under cool water to stop cooking. Drain well.

While the pasta cooks, warm 1/4 cup olive oil in a small skillet over medium-low heat. Add the sliced garlic and red pepper flakes if using. Cook gently until the garlic is just golden and fragrant, about 3-4 minutes. Do not let it brown or it will turn bitter. Remove from heat immediately.

Add the warm pasta to the bowl with the tomato mixture. Pour the garlic oil over everything, scraping in the garlic slices. Add the mozzarella pearls and grated Parmigiano-Reggiano. Toss thoroughly, adding a splash of reserved pasta water if the mixture seems dry. The residual heat from the pasta will slightly soften the mozzarella and help the dressing coat every piece.

Taste and adjust seasoning with more salt, pepper, or balsamic glaze as needed. Transfer to a serving platter, drizzle with a little more olive oil, and finish with flaky sea salt and extra grated cheese. Serve warm or at room temperature.