In a small saucepan over medium heat, combine 1 cup balsamic vinegar and 2 tablespoons honey. Bring to a gentle boil, then reduce heat and simmer for about 10-15 minutes, stirring occasionally, until the mixture thickens and coats the back of a spoon. Set aside and keep warm.
Pat the chicken breasts dry with paper towels. Season both sides generously with salt and pepper. Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Once hot, add the chicken breasts and sear for 5-6 minutes on each side, until golden brown and cooked through (internal temperature should reach 165°F/74°C). Remove chicken from the pan and set aside.
In the same skillet, add halved cherry tomatoes. Cook for 3-4 minutes, stirring occasionally, until tomatoes soften and release their juices. Season with a pinch of salt and pepper.
Return the chicken breasts to the skillet, placing them on top of the tomatoes. Layer slices of fresh mozzarella cheese over each breast. Cover the skillet with a lid and let the cheese melt for 2-3 minutes.
Remove the skillet from heat. Sprinkle fresh basil leaves over the chicken and tomatoes. Drizzle the balsamic-honey glaze generously on top. Serve immediately, spooning extra glaze and tomatoes over the chicken.
