Step 1

Slice half of the lemon into thin (⅛-inch) rounds; reserve the other half.

Step 2

Cut the chicken breasts in half horizontally to make 4 thin cutlets. Place each cutlet under plastic wrap and pound it until even and flat. Season the cutlets on both sides with 1 teaspoon of the salt and ½ teaspoon of the pepper.

Step 3

Place ¼ cup of the flour in a shallow dish. In another shallow dish, beat the eggs with the milk and ¼ teaspoon of the salt.

Step 4

In a large skillet, heat the oil over medium heat. Working in batches, dredge the chicken in the flour, shaking off the excess, then dip in the egg mixture, allowing the excess to drip off, and then immediately add it to the skillet. Cook the chicken until golden brown, 3 to 4 minutes per side; remove to a plate. Pour off the oil and wipe out the skillet.

Step 5

In the same skillet, cook the lemon slices over medium heat until fragrant and lightly golden, about 30 seconds. Remove to a plate. Add 2 tablespoons of the butter and the shallot and cook, stirring frequently, until the shallot is lightly golden, 1 to 2 minutes. Whisk in the remaining 2 tablespoons flour and cook, whisking frequently, until the flour turns lightly golden, 2 to 3 minutes. Whisk in the wine, then the chicken broth and bring to a simmer. Simmer, stirring frequently, until the sauce is thickened and coats the back of a spoon, 4 to 6 minutes. Squeeze in the juice from the remaining lemon half and stir in the remaining tablespoon of butter and ¼ teaspoon each salt and pepper. Add the chicken and lemon slices back to the skillet. Spoon the sauce over the chicken and continue cooking until the chicken is heated through. Sprinkle with parsley before serving with mashed potatoes, rice, pasta, or bread.