Optional but recommended: Prepare a brine with 4 cups cold water, 1/4 cup kosher salt, and 2 tablespoons sugar. Submerge chicken for 30 minutes. Rinse and pat dry completely.
Position oven rack in the middle and preheat to 425°F (220°C). Line baking dish with foil.
In a small bowl, whisk together olive oil, balsamic vinegar, minced garlic, Italian herbs, salt, and pepper. Coat chicken breasts evenly with this mixture.
Bake chicken at 425°F for 10 minutes to create an initial sear.
Without opening the oven, reduce temperature to 375°F (190°C) and continue baking for 5 minutes.
Remove chicken from oven and top each breast with alternating slices of tomato and mozzarella. Tuck basil leaves between some of the slices.
Return to oven and bake for an additional 7-10 minutes until internal temperature reaches 163°F (73°C) at the thickest part.
Remove from oven and let rest for 5 minutes to allow temperature to rise to 165°F and juices to redistribute.
Drizzle with balsamic glaze and extra virgin olive oil. Season with a pinch of salt and pepper, and garnish with additional fresh basil leaves.