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Combine the balsamic vinegar and brown sugar in a saucepan. Bring to a boil, then reduce to a simmer. Simmer until reduced by half; about 10-15 minutes, stirring several times. It should be thick enough to coat the back of a spoon.
In a medium bowl, gently combine the diced tomatoes, minced garlic, kosher salt, freshly ground black pepper, red wine vinegar, olive oil, and chopped basil. Taste and add more salt and black pepper if desired.
In a small bowl, combine 2 tablespoons of olive oil, Italian seasoning, kosher salt, and black pepper. Brush both sides of the chicken breasts with it.
Heat 1 1/2 to 2 tablespoons of olive oil in a large skillet over medium-high heat. Cook the chicken until browned on both sides and cooked through. Reduce heat to medium-low and top each chicken breast with about 1/3 cup of shredded mozzarella cheese. Cover the skillet with aluminum foil and let the cheese melt. Top with the tomato mixture and drizzle with the balsamic glaze. For best results, serve immediately.
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