Preheat the oven to 400 degrees . In a medium saucepan, bring the potatoes and enough cold water to cover 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, butter and lemon peel 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 one-quarter 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 plate. 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.