Arrange an oven rack in the upper third of the oven and heat the oven to broil.

Pat 1 ½ pounds chopped boneless, skinless chicken dry with paper towels. Season all over with ¾ teaspoon of the kosher salt and ¼ teaspoon black pepper.

Heat 1 tablespoon of the olive oil in a large oven-safe skillet over medium heat until shimmering. Add half of the chicken in a single layer. Cook until golden-brown on the bottom and the chicken easily releases from the bottom of the pan, 4 to 5 minutes. Flip the chicken and cook until golden brown on the second side, about 2 minutes more. Transfer to a plate (some pieces may not be cooked through).

Heat 1 ½ teaspoons of the olive oil in the skillet and repeat browning the remaining chicken. Transfer to the plate.

Heat the remaining 1 tablespoon olive oil in the skillet until shimmering. Add 2 cups grape tomatoes and the remaining ½ teaspoon kosher salt. Cook until barely starting to soften, just before the tomatoes split, about 1 minute.

Stir in the chicken and any accumulated juices on the plate, 2 tablespoons balsamic glaze, and 2 minced garlic cloves.

Remove the skillet from the heat. Scatter 6 ounces halved fresh mozzarella cheese balls over the chicken and tomatoes. Broil until the cheese is melted, 2 to 3 minutes. Drizzle with more balsamic glaze and garnish with chopped fresh basil leaves if desired.