In a large bowl or ziplock bag, add chicken, ½ cup balsamic, olive oil, italian seasoning, and salt and pepper to taste.
Heat a grill or saute pan to medium heat. Remove chicken from marinade (reserving the liquid) and place on the grill. Cook on each side for 4-5 minutes or until the internal temperature reaches 165°
While the chicken is cooking, pour the remaining marinade and ½ cup of balsamic in a small sauce pot over medium low heat. Bring to a simmer and reduce by half, until the vinegar becomes this and syrupy.
Brush the grilled chicken with the balsamic reduction and top each breast with equal amounts of mozzarella. Close the grill lid or cover the saute pan and turn off the heat, allowing the cheese to melt for 3-5 minutes.
While the cheese is melting, pit and cube one avocado and 2 seeded roma tomatoes into a bowl. Drizzle with olive oil, season with salt and pepper, and add in basil. Stir to combine.
To serve, place chicken on a bed of greens or cauliflower rice, top with heaping tablespoons of avocado salad, and drizzle with additional olive oil and sprinkle with basil.