Fill a pot with cold salted water, add potoates and bring to a boil for about 20 minutes or just until tender; drain and let cool, but do not peel. Cut in half and set aside.
Preheat the broiler. Cut a small slice off both ends of each lemon, then cut in half crosswise.
Place lemons, flesh side up, in a flameproof baking dish, brush with olive oil, and season with salt and pepper.
Broil lemons 6 inches or more from the heat until browned and soft, about 10 minutes. Let cool.
Squeeze the lemon halves over a sieve suspended over a bowl. Pushing the pulp through the sieve Discard the lemon shells.
Season the chicken with salt and pepper.
Heat the remaining 1/4 cup olive oil in a large oven proof saute pan 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 to a platter.
Return the pan to medium-high heat, add the potatoes, season with salt and pepper, and cook, stirring, until brown all over, about 5 minutes. Drain off the excess oil.
Arrange the chicken breasts on top of the potatoes and place in the oven to reheat and cook through, about 10 minutes.
When done, 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 and add the garlic. Sauti briefly until light brown. Immediately add the reserved roasted lemon juice (this final flash of heat will cook off any residual acid flavor), stock, rosemary, and parsley.
Stir and scrape up all the browned bits that cling to the bottom and sides of the pan.
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.