How to Make It
In a food processor, pulse the olives with the anchovies, apricots, garlic, capers and 2 tablespoons of the olive oil until the tapenade is a coarse puree.
Make a horizontal incision in each piece of chicken to form a pocket. Spoon the tapenade into each pocket and press gently to close. Season the chicken breasts on both sides with salt and pepper.
In a large skillet, melt 2 tablespoons of the butter in the remaining 1 tablespoon of olive oil. Add the chicken breasts, cover and cook over moderate heat, flipping once, until golden brown, about 6 minutes. Transfer the chicken to a plate and cover with aluminum foil.
In the same skillet, add the mushrooms and onion and cook over moderate heat, stirring occasionally, until the onion is translucent, about 2 minutes. Add the wine and cook, scraping up the brown bits in the pan, until the liquid is slightly reduced, about 2 minutes more. Return the chicken to the pan and simmer until warmed through, about 2 minutes. Add the remaining 2 tablespoons of butter to the sauce and season with salt and pepper. Transfer the chicken breasts to 4 plates and spoon the sauce over the top. Garnish with parsley and chives and serve.