Place one chicken breast between two sheets of plastic wrap or inside a large zip top bag. Using the flat side of a meat mallet, pound the chicken to an even thickness of about 1/4 to 1/2 inch.
Pat the cutlets dry with paper towels and season both sides generously with salt and pepper.
In a shallow dish, whisk together the flour and grated parmesan cheese until evenly combined. One at a time, place each chicken cutlet in the flour mixture and turn to coat both sides completely, shaking off any excess.
Heat the olive oil and 2 tablespoons of butter in a large skillet over medium high heat until the butter is melted and just beginning to foam. Add two of the chicken cutlets to the pan, being careful not to overcrowd the skillet.
Cook for 3 to 4 minutes on the first side, until deeply golden brown, then flip and cook another 3 to 4 minutes on the second side. The chicken should reach an internal temperature of 165°F.
Transfer the cooked cutlets to a plate and tent loosely with foil to keep warm. Repeat with the remaining chicken cutlets, adding a little more oil to the pan if needed.
Reduce the heat under the skillet to medium and add the minced shallot. Cook for about 1 minute, stirring often, until softened and fragrant. Add the garlic and cook for another 30 seconds, just until fragrant, taking care not to let it burn.
Pour in the white wine and use a wooden spoon to scrape up any browned bits stuck to the bottom of the pan. Let the wine simmer for 2 to 3 minutes, until it has reduced by about half.
Add the chicken broth and lemon juice to the pan and bring the mixture to a gentle simmer. Stir in the drained capers and let the sauce simmer for another 2 to 3 minutes, until it starts to thicken slightly.
Reduce the heat to low and whisk in the remaining 2 tablespoons of butter, a little at a time, until the sauce turns glossy. Taste the sauce and adjust the seasoning with a little extra salt, pepper, or lemon juice if needed.
Nestle the cooked chicken cutlets back into the skillet, spooning the sauce over the top of each piece. Let everything warm together for 1 to 2 minutes over low heat.
Sprinkle with the chopped parsley and arrange a few lemon slices over the top for garnish. Serve immediately while the sauce is warm and glossy.