Preheat the oven to 400 degrees F. Drizzle 1 tablespoon of oil onto a rimmed baking sheet and use your fingers to evenly spread it out and coat the bottom of the pan. Sprinkle with the Italian seasoning, 1 tablespoon salt and a few grinds of pepper. Scatter the cherry tomatoes on the baking sheet and shake it back and forth so they become evenly coated in the oil and seasoning. Arrange the chicken in the center of the baking sheet, pushing the tomatoes out to the perimeter, and drizzle the chicken with the remaining tablespoon of oil.
Bake until the tomatoes are soft, blistered and begin to burst, 20 to 25 minutes.
Meanwhile, add the vinegar to a small saucepan and bring to a boil over medium-high heat. Lower the heat to medium and simmer until the vinegar has reduced by half and is thick enough to coat back of the spoon, 7 to 8 minutes.
Remove the baking sheet from the oven and drape a slice of mozzarella over each chicken breast. Bake until the cheese melts and an instant-read thermometer inserted into the thickest part of the breast has reached 160 degrees F, 5 to 7 minutes (the chicken will carryover cook to 165 F).
Sprinkle the chicken and tomatoes with the basil and drizzle with the balsamic reduction. Spoon the collected pan juices over the tomatoes and chicken. Serve on a bed of arugula and with crusty bread to mop up the pan juices.