Cut each chicken breast in half horizontally, making 4 thinner cutlets total.
One at a time, place each cutlet between 2 pieces of plastic wrap; using the smooth side of a meat mallet, gently pound them to an even thickness of about ½ inch.
Season the chicken with salt and pepper.
Place the flour in a shallow bowl or on a plate; lightly dredge each piece of chicken in the flour, coating both sides and shaking off any excess.You probably won't use all of the flour; discard whatever is left.
In a large skillet (I use a 12-inch skillet), heat 2 tablespoons of the butter and the olive oil over medium-high heat.
Add the chicken to the skillet; cook until are nicely golden-brown (about 2 to 3 minutes per side).The chicken will not be cooked through at this point; we are just looking to get some color in this step. It will finish cooking in the sauce later.
Remove the chicken from the skillet to a plate; set aside.
Reduce the heat to medium-low. Add the shallots and garlic; cook, stirring constantly for 1 to 2 minutes just until the shallots begin to soften and the garlic is fragrant.Be very careful not to burn the garlic, as it will become bitter. Watch the heat and adjust it as needed. You could even remove the pan from the heat for a minute or two to cool it down if you need to.
Add the wine and deglaze the skillet by scraping up any browned bits from the bottom of the pan.
Increase the heat and bring to a simmer; simmer for 1 to 2 minutes or until the wine has reduced by about half.
Add the chicken broth, lemon juice and capers; bring back up to a simmer.
Return the chicken to the pan; gently simmer just until the chicken is cooked through (internal temperature of 165°F), about 4 to 5 minutes.
Add the remaining 2 tablespoons of butter to the pan and stir until it has melted into the sauce. Taste the sauce and add salt and pepper if needed.
To serve: spoon some of the sauce over each piece of chicken and sprinkle with chopped fresh parsley. Serve immediately.