In a large bowl combine the chicken breasts, minced garlic, 2 tablespoons balsamic vinegar, 2 tablespoons olive oil, Italian seasoning, 1/2 teaspoon salt, and 1/4 teaspoon black pepper; stir to coat the chicken. Cover and refrigerate to marinate for at least 30 minutes or up to 24 hours.
While the chicken marinates or just before cooking, make the avocado-tomato salsa: in a medium bowl add the halved tomatoes, diced avocados, chopped basil, 1 teaspoon olive oil, juice of 1/2 lemon, and a dash of salt and pepper; gently stir to combine and cover until ready to use.
Heat a large heavy-duty skillet or grill pan over medium-high heat until hot.
Add the marinated chicken and cook 8–10 minutes per side (depending on thickness) until the internal temperature reaches 165°F (74°C) and the chicken is cooked through.
Top each cooked chicken breast with a slice of fresh mozzarella; allow the cheese to melt slightly in the pan for 1–2 minutes.
Spoon the avocado-tomato salsa over the melted mozzarella on each chicken breast and serve immediately.