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Preheat the oven to 400 degrees F (200 degrees C). Line a rimmed 13x18-inch baking sheet with foil.
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Sprinkle chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper.
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Combine flour and Parmesan cheese in a shallow dish. Dredge chicken in flour mixture to coat, shaking off excess.
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Slice one of the lemons. Juice the remaining 2 lemons, you should have about 6 tablespoons of juice.
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Whisk together 3/4 cup chicken broth, oil, lemon juice, smoked paprika, and oregano in a small bowl. Add potatoes to the prepared baking sheet. Sprinkle with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper.
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Pour sauce mixture over the potatoes. Bake in the preheated oven for 10 minutes. Remove from oven.
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Nestle chicken in between potatoes, turning to coat in the sauce mixture, adding additional broth to the baking sheet, if needed.
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Return to oven and bake 30 minutes, until chicken is done (165 degrees F or 74 degrees C), and potatoes are tender. In the last 10 minutes of bake time, top chicken and potatoes with sliced lemon, capers, and garlic.
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Garnish with parsley.