Marinate Chicken: In a medium bowl, whisk together the balsamic vinegar, 1 tablespoon of avocado oil, Dijon mustard, Italian seasoning, ½ teaspoon salt, ¼ teaspoon pepper, and minced garlic. Add the chicken and toss to coat. Cover with plastic wrap and refrigerate for at least 30 minutes, up to 2 hours.
Sear Chicken: Preheat oven to 375°F (190°C). Heat a large skillet over medium high heat. Add the remaining 1 tablespoon of avocado oil. Remove chicken from marinade, wiping off excess. Place chicken in the pan and sear for 3 minutes per side, until golden brown.
Assemble Melts: Place the ciabatta rolls, cut side up, on a small baking sheet. Place the seared chicken on one half of each roll. Top each with 4 slices of mozzarella cheese.
Bake: Transfer the baking sheet to the oven and bake for 8-10 minutes, until the chicken reaches 165°F (74°C) and the cheese is melted. Optionally, switch to broil for the last 2 minutes for a golden top.
Prep Tomatoes: While the melts bake, place tomato slices on a paper towel lined plate. Sprinkle with salt and pepper. Let sit for a few minutes, then pat dry to remove excess moisture.
Make Green Goddess Sauce: In a small food processor or blender, combine the Greek yogurt, mayonnaise, basil, parsley, green onions, garlic clove, and lemon juice. Blend until smooth. Season with salt and pepper to taste.
Assemble Sandwiches: Spread a thick layer of the green goddess sauce on the empty side of each roll. Top with the seasoned tomato slices and half of the arugula. Close the sandwiches with the chicken mozzarella side.
Serve: Slice each sandwich in half with a serrated knife and serve immediately.