Cut the chicken breasts in half horizontally to make 8 thin-cut cutlets.
Mix together the flour, salt, pepper, and grated parmesan in a shallow bowl. Dredge the chicken cutlets in the flour mixture so until well coated.
Heat 2 tablespoons olive oil and 2 tablespoons of butter in a large skillet over medium heat. Once melted, add half of the coated chicken cutlets. (You don't want to overcrowd the pan.) Cook for about 3 minutes per side until golden brown and cooked through.
Remove the cooked chicken to a plate. Add another 2 tablespoons of olive oil and butter to the skillet and repeat with the remaining chicken. Cover the plate with foil and keep the chicken warm in the oven while you make the sauce.
The browned bits at the bottom of the skillet should be golden to medium in color. Leave them in the pan for extra flavor, along with any remaining chicken juices and butter/oil in the skillet. If the brown bits at the bottom of the skillet are very dark, clean it out. Then add 1 tablespoon each of olive oil and butter to the skillet.
With the skillet over medium heat, add the chicken stock (or wine), lemon juice, garlic, and capers to the pan. Use a wooden spoon to scrape up the browned bits (if needed). Simmer until the sauce has reduced by half. Whisk in the remaining 2 tablespoons of butter.
Plate the chicken over cooked pasta with some of the sauce spooned over top. Add a couple of lemon slices for garnish before serving warm.
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Notes
The extra 1 tablespoon of butter and oil is not included in the nutritional information, as it's not always necessary to clean the pan.
You can add upto another 1/4 cup of capers to your taste preference.