Prep the chicken: Pat the chicken dry. Season both sides with salt and pepper. In a shallow bowl, whisk the egg and milk. In another bowl, mix breadcrumbs and Parmesan.

Bread the cutlets: Dip each chicken breast in the egg mixture, letting the excess drip off. Coat in the breadcrumb mixture, pressing to adhere. Set on a plate.

Brown the chicken: Heat 2 tablespoons olive oil and 1 tablespoon butter in a large skillet over medium heat. Add chicken in batches. Cook 3–4 minutes per side until golden and almost cooked through. Transfer to a plate. Don’t overcrowd the pan.

Sauté the prosciutto and mushrooms: Add remaining 1 tablespoon olive oil to the skillet. Stir in prosciutto and cook 1 minute until edges crisp. Add mushrooms and a pinch of salt. Cook 5–6 minutes until browned and their moisture cooks off.

Add aromatics: Stir in garlic and cook 30 seconds until fragrant. Scrape up browned bits with a wooden spoon.

Deglaze and simmer: Pour in white wine. Simmer 2–3 minutes until reduced by about half. Add chicken broth, lemon zest, and lemon juice. Bring to a gentle simmer.

Return chicken to the pan: Nestle the chicken into the sauce. Sprinkle in capers if using. Simmer 4–6 minutes, spooning sauce over the top, until chicken is cooked through and the sauce slightly thickens.

Add cheese: Sprinkle shredded cheese evenly over each piece of chicken. Cover the skillet, reduce heat to low, and cook 2–3 minutes until the cheese melts. If your skillet is oven-safe, you can broil 1–2 minutes to lightly brown the cheese. Watch closely.

Finish and serve: Stir in the remaining 1 tablespoon butter to silk up the sauce. Taste and adjust salt and pepper. Scatter parsley on top. Serve with extra sauce spooned over.