Slice the chicken breast in half through the center to form 4 thinner filets. Place the chicken pieces between 2 sheets of plastic wrap. Using the flat face of a meat mallet or a rolling pin, pound the chicken breasts to 1/4-inch thickness. Place the filets on a platter and season both sides with salt and pepper. Set aside to come to room temperature.

In a large cast-iron or heavy-bottomed skillet over medium-high heat, warm the olive oil and 3 tablespoons of the butter. When the butter has melted and starts to foam, add 2 pieces of chicken to the pan and cook until golden brown, approximately 2 to 3 minutes per side. Turn the heat down if the butter starts to brown. Transfer the cooked chicken to a clean platter and repeat with the other 2 paillards. Make sure the chicken is at 165°F when checked with a meat thermometer.

Reduce the heat of the pan to medium. Add the shallots to the pan and cook until softened, approximately 2-1/2 minutes. Add the garlic and continue to cook until it is aromatic, about 1 minute longer.

Raise the heat back up to medium-high. Stir the lemon juice into the pan. Add the chicken broth and stir with a wooden spoon, scraping up any browned bits. Simmer until the sauce reduces by half. This should take about 3-1/2 minutes.

Add the remaining 3 tablespoons of butter to the sauce. As soon as the butter melts, add the lemon zest and chives to the pan. Check the seasoning and add salt and pepper if desired.

Return the chicken to the pan, including any juices that are on the plate. Heat while spooning sauce over the paillards until just warmed through.

Serve the chicken paillard with the sauce. Enjoy with a leafy green salad and a nice glass of wine.