Pat chicken breasts dry with paper towels. If needed, pound thicker ends to create even thickness so they cook uniformly. Brush both sides with 1 tablespoon olive oil, then season evenly with garlic powder, kosher salt, and black pepper.

Slice the mozzarella into 8 rounds about 1/4-inch thick. Halve the cherry tomatoes and tear any large basil leaves. Set aside at room temperature so the cheese melts better.

Preheat your grill or grill pan to medium-high heat, about 400 degrees F. Grill chicken for 5-6 minutes per side with the lid closed, until grill marks appear and the internal temperature reaches 155 degrees F. The chicken will still look slightly pink in the center - this is intentional.

Transfer chicken to a cooler side of the grill or reduce heat to medium. Top each breast with 2 slices of mozzarella, close the lid, and cook for 2-3 minutes until the cheese is melted and beginning to ooze. The chicken should now read 165 degrees F in the thickest part.

Transfer chicken to a serving platter. Immediately scatter the halved tomatoes and torn basil over the top. Drizzle with remaining 1 tablespoon olive oil and the balsamic glaze. Finish with a pinch of flaky sea salt and serve right away while the cheese is still warm and stretchy.