Prepare the vinaigrette by combining tablespoons of olive oil and other ingredients on a glass jar with a tight-fitting lid. Cover and shake vigorously until thoroughly combined. Set aside.
Place the chicken breasts on a large sheet of plastic wrap or wax paper spread out on a clean work surface. On a cutting board, cut the breasts in half horizontally (with the blade parallel to the counter). Arrange both halves on the sheets of plastic wrap or wax paper in a single layer.
To adjust the size of thick slices, cover the mat with a second sheet of plastic wrap or wax paper and pound to 1/4” thickness with a meat pounder or rolling pin. Remove the top cover and season the chicken cutlets on both sides with the garlic powder, teaspoon salt, and fresh ground pepper.
Add a drizzle of olive oil to a large nonstick skillet set over medium-high heat. Once hot, place chicken in and most of the lemon slices. Reserve 3 or 4 strips of lemon zest.
Cook until the chicken is golden brown on the bottom and releases easily from the skillet, approximately 3-4 minutes. Turn and cook on the remaining side another 2-3 minutes, or just until the chicken is cooked through.
Combine arugula, tomatoes, fresh parsley, and Parmesan cheese in a large bowl and drizzle 2-3 tablespoons of the vinaigrette on top. Season with salt and black pepper, to taste, and toss to combine.
To serve, divide the salad between 4 serving bowls or a large plate and top each with pieces of chicken and warm lemon wedges. Place the remaining dressing in a small bowl and offer it as a side while serving. Enjoy!