In a small saucepan, combine ⅓ cup of the vinegar and 1 tablespoon of the honey. Gently simmer over medium-low heat, stirring occasionally, until thickened by a little more than half, about 15 minutes. Remove from the heat and set aside.

While the balsamic reduces, arrange the chicken on a cutting board and stretch a large sheet of plastic wrap over the top. Lightly pound the chicken breasts into an even ½-inch thickness. Discard the plastic, then sprinkle the chicken with ½ teaspoon of the salt and ¼ teaspoon of the pepper, seasoning both sides.

In a large, heavy-bottomed skillet, heat 1 tablespoon of the oil over medium-high heat. Once the oil is hot but not smoking, swirl to coat the pan, then add the chicken breasts top-sides down, being careful of oil that may splatter. Let cook undisturbed for 4 minutes, or until lightly browned. Flip and cook an additional 3 minutes. Check for doneness. The chicken should reach 165 degrees F at the thickest part when tested with an instant-read thermometer, and when sliced, the juices should run clear. If it needs additional time, flip again and continue cooking for a few minutes on each side, until done. The total cooking time will vary depending upon the size of your chicken. Remove to a plate and cover to keep warm.

Without wiping out the pan, reduce the skillet heat to medium. Add the remaining 1 tablespoon oil, swirl to coat the pan, then add the red onion and let cook until slightly softened, about 3 minutes, scraping the bottom and sides of the pan as it cooks. Add the garlic and cook for 1 to 2 minutes until fragrant. Stir in the tomatoes, the remaining 1 teaspoon honey, ¼ teaspoon salt, and the thyme. Reduce the heat to medium-low and simmer, stirring occasionally, until the tomatoes begin to soften, about 5 minutes.

Return the chicken to the pan, nestling it into the tomatoes, and scatter the mozzarella over the top and around the chicken. Cover the pan and remove from the heat. Let stand for 1 to 2 minutes until the cheese becomes soft and melts. Uncover, drizzle with the balsamic reduction over the top, and sprinkle with the fresh basil. Serve warm.