Preheat the oven to 450 degrees. 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.
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 olive oil. Add the chicken, turn to coat. Cover bowl and refrigerate for 1-2 hours.
Heat a large cast-iron or oven-proof skillet over medium-high heat. Add the chicken, skin-side down, cover and cook until the skin browns, about 5 minutes. Remove chicken from pan; add the mushrooms and potatoes to the skillet, place chicken over mushrooms and potatoes and drizzle with any marinade remaining in the bowl and 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 is crisp, about 20 to 25 minutes.