Place chicken breasts in a large freezer bag. Use a rolling pin, mason jar, or the flat side of a meat tenderizer, pound the thickest part of the chicken breast until they are about 3/4" thick and even with the rest of the chicken.
Open the bag and add the oil, lemon juice, and Italian seasoning. Seal then roll the bag around gently to combine. Marinate in the fridge for at least 2, or up to 24 hours.
When you are ready to cook the chicken, preheat the grill to medium high heat.
Remove the chicken from the marinade and place on the preheated grill. Cook for about 5-7 minutes, or until the chicken gets golden brown and is easy to lift (if the chicken sticks, let it cook longer). Flip the chicken over using tongs and continue to cook until almost golden brown—about 9-10 minutes total.
Once the chicken is almost completely cooked, and has almost reached an internal temperature of 165 degrees F, add 3-4 slices of avocado, 2-3 slices of tomato, and 2 slices of pepper jack cheese. Close the grill for 2 more minutes or until the cheese is melted.
Remove cooked chicken from the grill and cover with foil for 5 minutes while the chicken breast cools down a little. Serve warm and enjoy!