In a medium saucepan, add the potatoes and enough cold water to cover, and bring to a boil. Lower the heat and simmer until the potatoes are fork-tender, about 15 minutes. Drain, then return the potatoes to the pot. Add the milk, 2 tablespoons of butter and lemon zest and mash until smooth. Season with salt and pepper. Transfer the potatoes to a work surface, flatten into an even round and cut into quarters.
Run your fingers under the skin of each chicken breast to separate it from the meat. Place 1/4 of the potatoes under the skin of each breast, gently pressing to evenly distribute. Place the stuffed chicken in a roasting pan, drizzle with the olive oil and season with salt and pepper. Bake until tender and the juices run clear, about 40 minutes. Transfer the chicken to a platter. Stir the lemon juice, parsley and remaining 2 tablespoons butter into the pan juices, scraping up any browned bits from the bottom. Spoon the sauce over the chicken and serve.