Working with one chicken breast at a time, place chicken on a cutting board. Hold it steady with the palm of your non-cutting hand and use a sharp knife to slice horizontally into two even cutlets. Repeat with remaining breasts. Cover cutlets with plastic wrap and use a meat mallet or rolling pin to pound them to about ¼-inch thickness.

Season chicken evenly on both sides with salt and pepper and let sit at room temperature for 15 minutes.

While chicken sits, place 1 cup (4 ½ ounces; 128 g) flour in a shallow dish. In a second shallow dish, beat eggs with milk.

Fill a large stainless steel skillet with about ¼ inch of oil and heat over high heat until shimmering. Working with two cutlets at a time, lightly dredge chicken in flour, shaking off excess, then dip in egg mixture, allowing excess to drip off. Carefully place chicken in hot oil and cook, undisturbed, until lightly browned and nearly cooked through, about 3 minutes. Flip and cook until lightly browned on the second side and an instant-read thermometer inserted into the thickest part registers 160°F (71°C), about 2 minutes. Transfer to a paper towel-lined plate and season with salt to taste. Repeat with remaining chicken, adding more oil as needed.

When all chicken is cooked, carefully discard frying oil and wipe the skillet clean. Return skillet to medium heat and add 2 tablespoons (28 g) butter. After butter is melted, add lemon slices and cook until lightly golden, about 30 seconds per side; transfer to a plate.

Add shallots to remaining butter in skillet and cook over medium heat, stirring frequently, until translucent, about 2 minutes. Whisk in remaining 1 ½ tablespoons (12 g) flour and cook, whisking constantly, until lightly golden and aromatic, about 2 minutes.

Gradually whisk in broth and wine. Bring to a gentle simmer and cook until sauce is lightly thickened, about 5 minutes. Off-heat, whisk in lemon juice, and chilled butter. Season to taste with salt and pepper. Return chicken to skillet and spoon sauce over cutlets. Cook over medium-low heat until chicken is warmed through, about 2 minutes. Top with reserved lemon slices and parsley. Serve.