Heat the avocado oil in a large pan over high heat. Season the chicken with salt and pepper. When hot, add the seasoned chicken to the pan and sear, browning over high heat, about 4 minutes per side, or until cooked through. Transfer the chicken to an oven-safe dish (or leave them in your pan if it is oven-safe).
Top with pesto, then alternating slices of tomato and mozzarella. Broil for 4 minutes, or until the cheese is melted. Drizzle balsamic over before serving.