• Adjust oven rack to middle position; heat oven to 200°F. Set wire rack on rimmed baking sheet and place sheet in oven.

To Make the Sauce:

• Heat 1 tablespoon butter in medium nonreactive saucepan over medium heat. When foaming subsides, add onion and cook, stirring occasionally, until translucent, 2 to 3 minutes.
  1 very small onion
• Add flour and stir until light golden brown, about 1 minute. Whisk in wine, lemon juice, and broth; increase heat to high and bring to boil, whisking constantly. Lower heat to medium-high and cook, whisking occasionally, until mixture is reduced to 1 ½ cups, 10 to 15 minutes.
  1 tablespoon all purpose flour, ½ cup dry white wine or vermouth, ⅓ cup lemon juice from 2 lemons, 2 ¼ cups low sodium chicken broth
• Strain sauce through mesh strainer, return to saucepan, and set aside.

To Make the Chicken:

• Set second wire rack on second rimmed baking sheet on counter. Whisk together flour, 1 teaspoon salt, and ¼ teaspoon pepper in pie plate.
  1 cup all purpose flour
• In second pie plate, whisk eggs and milk until combined. Season both sides of each cutlet with salt and pepper. Using tongs and working with 2 cutlets at a time, coat cutlets in seasoned flour; shake off excess flour.
  2 large eggs, 2 tablespoons milk
• Transfer cutlets to egg mixture; coat evenly and let excess run off. Return cutlets to seasoned flour; coat evenly and shake off excess flour. Place coated cutlets on wire rack on counter.
• Heat 1 tablespoon each butter and oil in 12-inch nonstick skillet over medium-high heat; when foaming subsides, place 4 cutlets in skillet. Cook until well browned, 1 ½ to 2 minutes. Carefully flip cutlets and continue to cook until lightly browned on second sides, 30 to 60 seconds.
• Transfer chicken to wire rack in oven. Wipe out skillet with paper towels. Repeat, using remaining 1 tablespoon each butter and oil to cook remaining cutlets in now-empty skillet. After transferring chicken to oven, wipe out skillet with paper towels.

To Finish the Sauce:

• Transfer the sauce to now-empty skillet and set over low heat; cook until the sauce is heated through, about 1 minute. Whisk in the remaining 2 tablespoons of butter; add salt and pepper to taste.

To Serve:

• Remove baking sheet with chicken from the oven; transfer 4 cutlets to the skillet, turn to coat with sauce, then transfer each serving (2 cutlets) to individual plates. Repeat with the remaining cutlets.
• Spoon 2 tablespoons additional sauce over each serving and sprinkle with parsley. Serve immediately, passing extra sauce separately.
  2 tablespoon minced fresh parsley leaves