Place pork chops between 2 sheets of parchment paper or plastic wrap. Pound with a meat mallet to 1/4-inch thickness. Sprinkle pork evenly with salt and pepper.

Place flour in a shallow dish; dredge seasoned cutlets in flour, and shake off excess. Heat 1 tablespoon each of butter and oil in a large nonstick skillet over medium-high until butter foam subsides. Add 2 pork cutlets, and cook until golden on bottom, 2 to 3 minutes per side. Transfer cutlets to a large plate; wipe out skillet.

Repeat cooking process with 1 tablespoon of the butter, remaining 1 tablespoon oil, and remaining 2 cutlets. Wipe out skillet.

Heat 1 tablespoon of the butter in skillet over medium. Add shallot and garlic; cook, stirring constantly, until fragrant and garlic just starts to brown, about 1 minute.

Add wine, and cook, stirring occasionally, until almost evaporated, about 1 minute. Add broth and lemon slices, and bring to a boil over medium; cook, stirring occasionally, until mixture is reduced by half, 3 to 4 minutes. Whisk in capers.

Reduce heat to low, and whisk in remaining 5 tablespoons butter, 1 to 2 pieces at a time, until melted and sauce is creamy, about 2 minutes.