Preheat the grill to medium-high heat and clean and lightly oil the grates.
Combine the diced fresh mozzarella, halved cherry tomatoes, chopped basil, and balsamic glaze in a medium bowl. Season lightly with salt and black pepper.
Place each chicken breast on a cutting board. Using a sharp knife, carefully cut a deep pocket horizontally into the thickest side without slicing completely through the chicken.
Divide the Caprese mixture between the chicken breasts and gently press the filling into each pocket. Secure the openings with toothpicks if necessary.
Brush the outside of each chicken breast lightly with olive oil and season with salt and pepper.
Place the Fresh Caprese Stuffed Grilled Chicken on the preheated grill. Grill for about 6–7 minutes per side, adjusting the cooking time according to thickness.
Cook until the thickest part of the chicken reaches a safe internal temperature of 165°F (74°C). Check the chicken itself rather than the mozzarella filling when taking the temperature.
Transfer the grilled chicken to a clean plate and let it rest for about 5 minutes. Remove any toothpicks before serving and finish with extra basil or balsamic glaze if desired.