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

2. FOR 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. 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 1/2 cups, 10 to 15 minutes. Strain sauce through mesh strainer, return to saucepan, and set aside.

3. FOR THE CHICKEN: Set second wire rack on second rimmed baking sheet on counter. Whisk together flour, 1 teaspoon salt, and 1/4 teaspoon pepper in pie plate. 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. 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.

4. 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 1/2 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.

5. TO FINISH SAUCE AND SERVE: Transfer sauce to now-empty skillet and set over low heat; cook until sauce is heated through, about 1 minute. Whisk in remaining 2 tablespoons butter; adjust seasoning with salt and pepper. Remove baking sheet with chicken from oven; transfer 4 cutlets to skillet, turn to coat with sauce, then transfer each serving (2 cutlets) to individual plates. Repeat with remaining cutlets. Spoon 2 tablespoons additional sauce over each serving and sprinkle with parsley. Serve immediately passing extra sauce separately.

Serves 4