Preheat the broiler.
Cut a small slice off both ends of each lemon, and then cut in half crosswise. Arrange the lemons, flesh side up, in a flameproof nonreactive baking dish, and season with salt. Broil 6 inches or more from the heat until browned and soft, about 10 minutes. Cool slightly. Squeeze the lemon halves over a sieve suspended over a bowl. Push and stir the pulp through the sieve with a rubber spatula or wooden spoon to get as much juice out as possible. Discard the lemon peels.
Reduce oven temperature to 425 degrees F.
Season chicken with salt and pepper. Heat the olive oil in a large ovenproof skillet over medium-high heat until hot. Add the chicken, lower the heat to medium, and cook, turning once, until brown on both sides, about 5 minutes. Remove chicken to a plate or baking sheet.
Return the pan to medium-high heat, add the potatoes, season with salt and pepper, and cook, stirring and tossing, until brown all over, about 10 minutes. Drain off the excess oil. Arrange the chicken breasts on top of the potatoes and place in the oven until potatoes and chicken are cooked through, about 10 to 15 minutes. Remove the chicken to a platter and put the pan with the potatoes over medium-high heat. Toss well so the pan juices are absorbed into the potatoes. Scrape the potatoes out of the pan onto the platter around the chicken.
Return the pan to medium-high heat, add the garlic and rosemary, and cook until fragrant, about 1 minute. Add 1 cup of chicken broth and reserved lemon juice. Stir and scrape up all the browned bits that cling to the bottom and sides of the pan, and then add the remaining 1 cup broth. Bring to a simmer stir in parsley and season to taste with salt and pepper. If the sauce tastes too lemony, stir in the optional butter. Pour the sauce over the chicken and potatoes and serve immediately, garnished with rosemary sprigs
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