Slice chicken breasts in half length-wise (if extra large). Lightly pound to an even thickness, about a ½ inch. Generously season both sides of the cutlets with salt.
Place flour in a shallow bowl, lightly dredge chicken in the flour, shaking off the excess.
Cook the cutlets in two or three batches. Add olive oil and a teaspoon or two of butter to a skillet. Sear the cutlets over med-high heat for about 3 minutes, turn them over for another 30 seconds to a minute. The chicken will get heated and finish cooking in the sauce, so we are just going for a light sear. Add more oil and butter if needed for each batch. Remove chicken to a plate and set aside.
In the same pan, without cleaning it, sauté the shallots and capers for 3 minutes. Add garlic, sautéing another minute. Add wine and scrape all the brown bits off the bottom of the pan, simmer for 3 minutes.
Add chicken broth and black pepper, heat to a simmer and whisk in butter until emulsified. Add chicken and heat 2-3 minutes until chicken is warm and cooked through.
Add lemon juice, parsley and adjust salt and pepper to taste.
