Place the chicken breasts on a cutting board and pat them dry with paper towels. Place a sheet of stretch film over the chicken and pound it to ¾-inch thickness. Repeat the same process for the rest of the chicken breasts and set them aside.
To make the marinade, in a large bowl, whisk together ½ cup of balsamic vinegar, ¼ cup of olive oil, 1 tablespoon of honey, 1 teaspoon of Dijon mustard, 2 cloves of minced garlic, 1 teaspoon of kosher salt, and ½ teaspoon of black pepper.
Add the chicken breast to the bowl, ensuring it is fully submerged in the liquid. Cover with plastic wrap and let them sit on the counter while you work on the garnish and preheat the grill.*
To make the caprese garnish, place 1 cup of halved cherry tomatoes, 1 clove of minced garlic, 1 tablespoon of olive oil, 1 tablespoon of balsamic vinegar, ½ teaspoon of kosher salt, and ¼ teaspoon of black pepper in a bowl. Toss to combine. Add the torn basil leaves and set them aside.
When ready to cook, preheat the grill over medium-high heat, about 450°F (232°C). Once hot, scrape the grates to remove any food residue and oil the grates.
Remove the chicken breasts from the marinade, letting the excess drip off, and place them on the hot grill. Grill for about 4-5 minutes, flip, and then cook for another 3-4 minutes. Check with a meat thermometer to ensure that the internal temperature registers somewhere between 160-165°F (71-74°C).
Place the mozzarella slices (2 slices on each chicken) over the chicken breasts, turn the heat to medium-low, cover the grill, and continue to cook until the cheese is slightly melted for about 3-4 minutes.
If your grill has a wire shelf on top (inside the lid), place the chicken on it to prevent further cooking (and potentially drying).
Transfer the chicken breasts to a large platter and cover loosely with foil. Let them rest for 5-10 minutes.
Right before serving, garnish the chicken with the tomato salad we made earlier.
If using, drizzle it with balsamic glaze and serve while still warm.