Sprinkle chicken on both sides with saltand pepper. Fold wings underneath to secure them. Melt butter in a cast-iron skillet large enough to hold the flattened chicken comfortably. Put chicken in, skin-side down, cover chicken firmly with a plate and put about 5 lbs of weight on top of the plate. Cook over low heat until skin is nicely browned, about 25 min. Turn chicken skin-side up, replace weight, cook about 15 more min. Remove chicken and pour off fat, reserving 2 Tbsp. Add flour to fat and whisk. Gradually add chicken broth and cook until thickened. Return chicken (skin-side up), add salt and pepper, replace plate and weight, and continue cooking over low heat about 30 min, or until chicken is exceptionally tender. Spoon the gravy over the chicken, cut into serving pieces, and serve with fluffy rice.