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Preheat the oven to 400 degrees F. Line a sheet pan with parchment paper.
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Place each chicken breast between 2 sheets of parchment paper or plastic wrap and pound out to ¼-inch thick. Sprinkle both sides with salt and pepper and dried oregano.
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Mix the flour, ½ teaspoon salt, and ¼ teaspoon of pepper in a shallow plate. In a second plate, beat the egg and ½ tablespoon of water together. Place the bread crumbs on a third plate. Dip each chicken breast first in the flour, shake off the excess, and then dip in the egg and bread crumb mixtures.
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Heat 1 tablespoon of olive oil in a large saute pan over medium to medium-low heat. Add the chicken breasts and cook for 2 minutes on each side, until browned. Place them on the sheet pan and allow them to bake for 5 to 10 minutes while you make the sauce.
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For the sauce, wipe out the saute pan with a dry paper towel. Over medium heat, melt 1 tablespoon of the butter and then add the lemon juice, wine, ½ teaspoon salt, and ¼ teaspoon pepper. Boil over high heat until reduced in half, about 2 minutes. Off the heat, add the remaining 2 tablespoons of butter and the capers, and swirl to combine. Pour the sauce over the chicken breasts.