1/2 cup vegetable oil
1 cup all-purpose flour
3/4 teaspoon salt
1/2 teaspoon black pepper
3 large eggs
1/2 stick (1/4 cup) unsalted butter
1/2 cup dry white wine
1/2 cup low-sodium chicken broth
3 tablespoons fresh lemon juice plus 1 whole lemon, thinly sliced
1/4 cup chopped fresh flat-leaf parsley
Place chicken breasts between 2 sheets of plastic wrap and gently pound chicken with flat side of a meat pounder or with a rolling pin until 1/4 inch thick.
Heat oil in a 12-inch heavy skillet over moderate heat until hot but not smoking.
While oil is heating, stir together flour, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a shallow bowl. Dredge 2 pieces of chicken, 1 piece at a time, in flour mixture, shaking off excess. Lightly beat eggs in another shallow bowl. When oil is hot, dip floured chicken into beaten eggs to coat, letting excess drip off, then fry, turning over once, until golden brown and just cooked through, about 4 minutes total. Transfer to a plate lined with paper towels and keep warm, loosely covered with foil. Fry remaining chicken in same manner.
Pour off and discard oil, then wipe skillet clean and heat butter over low heat until foam subsides. Add wine, broth, and lemon juice and boil, uncovered, stirring occasionally, until sauce is reduced to about 1/2 cup, about 6 minutes. Stir in parsley and remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Spoon sauce over chicken and top with lemon slices.