Cover the potatoes with cold water in a saucepan and salt the water. Bring to a boil over medium-high heat and cook until tender, about 8 minutes; drain and set aside.
While the potatoes are cooking, preheat the oven to 450F. Pile the rosemary leaves, garlic, 2 teaspoons salt and the red pepper flakes on a cutting board, then mince and mash into a paste using a large knife. Transfer the paste to a bowl. Stir in the juice of 1 lemon and the olive oil. Add the chicken and turn to coat.
Heat a large cast-iron skillet over medium-high heat. Add the chicken, skin-side down, cover and cook until the skin browns, about 5 minutes. Turn the chicken; add the mushrooms and potatoes to the skilled and drizzle with the juice of the remaining lemon. Add the rosemary sprigs and the squeezed lemon halves to the skillet; transfer to the oven and roast, uncovered until the chicken is cooked through and the skin in crisp, 20 to 25 minutes.
