Assemble all necessary ingredients.

Lay each pork chop between two sheets of parchment paper or plastic wrap. Use a meat mallet to gently pound the chops to a uniform thickness of ¼ inch. Season the meat evenly with salt and black pepper.

Place the flour in a shallow dish. Dredge each seasoned pork chop in the flour, ensuring an even coating, and shake off any excess. Heat 1 tablespoon of butter along with 1 tablespoon of olive oil in a large nonstick skillet over medium-high heat until the butter stops foaming. Sear two pork chops at a time, cooking each side for 2 to 3 minutes until golden brown. Transfer the cooked chops to a large plate and clean the skillet. Repeat the process with another tablespoon of butter and the remaining olive oil for the other two chops, then wipe out the skillet once more.

Melt an additional tablespoon of butter in the same skillet over medium heat. Add the minced shallots and sliced garlic, stirring continuously for about one minute until aromatic and the garlic begins to lightly brown. Deglaze the pan with white wine, allowing it to cook for approximately one minute until it nearly evaporates. Introduce the chicken stock and lemon slices, bringing the mixture to a boil. Simmer for 3 to 4 minutes, stirring occasionally, until the liquid reduces by half. Stir in the capers.

Reduce the heat to low and incorporate the remaining butter into the sauce, adding one or two pieces at a time. Whisk gently until the butter is fully melted, creating a smooth and creamy sauce.

Arrange the cooked pork chops on serving plates. Generously spoon the prepared sauce over the meat and garnish with freshly chopped parsley. Serve immediately.