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Prepare the vinaigrette by combining all ingredients in a glass jar with a tight-fitting lid. Cover and shake vigorously until thoroughly combined. Set aside.
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Place the chicken on a large sheet of plastic wrap or wax paper spread out on a clean work surface. Cut the breasts in half horizontally (with the blade parallel to the counter) and arrange both halves on the plastic wrap or wax paper in a single layer.
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Cover with a second sheet of plastic wrap or wax paper and pound to ¼” thickness with a meat mallet or rolling pin. Remove the top cover and season on both sides with the garlic powder, salt, and black pepper.
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Add the olive oil to a large skillet set over medium-high heat. Once hot, add the chicken and most of the lemon slices, reserving 3 or 4 slices for garnish.
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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.
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Note: Squeeze the lemon slices with tongs while the chicken cooks to release the juice.
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Remove from heat and discard the lemon rinds. Transfer the chicken to a large platter and place the reserved lemons in the skillet for approximately 30 seconds per side to warm up. Place the warm lemons on the platter with the chicken and set aside.
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Add the arugula, tomatoes, fresh parsley, and Parmesan cheese to 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.
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To serve, divide the salad between 4 serving bowls and top each with some of the chicken and warm lemon slices. Offer remaining dressing on the side while serving. Enjoy!