Add the vinegar to a small saucepan and bring to a boil over medium heat. Continue to boil the vinegar until reduced to approximately 1/3 of a cup. The vinegar should be syrupy at this point.

Allow the reduction to cool to room temperature. Transfer to an airtight container and store until ready to use.

Lay the chicken breasts on a flat surface and, using a sharp knife, cut a small pocket into the side of the breast lengthwise being sure not to cut all the way through. 

Spread 1 tablespoon basil pesto inside the pocket. Stuff two slices of mozzarella and two slices of tomato inside the chicken. Close the open side of the chicken breast with a small skewer. Repeat with the remaining chicken.

Heat the grill. Place the chicken breast on the grill and cook for 6-7 minutes. Flip the chicken breasts over and continue cooking for an additional 7 minutes or until cooked through. (Some cheese will ooze out but that's totally fine.)

Transfer the chicken to a platter and allow to rest for 5 minutes.

Drizzle the balsamic glaze over the chicken and serve immediately.