Trim chicken pieces of any excess fat, then rinse and pat dry with paper towels.
Heat about 1/2 inch of oil in a deep-fryer, a large, deep skillet or a broad saucepan with a cover. While oil is heating, season chicken with 1/2 teaspoon of the salt. In a plastic bag, mix together flour, remaining tablespoon salt, chili powder, cumin, black pepper and cayenne. Toss the chicken in the bag, 2 or 3 pieces at a time, until they are well coated with flour. Put them on a wire rack as you finish.
When the oil reaches 350 degree F (a cube of bread crisps up golden brown in 45 seconds), raise the heat to high and begin to slowly but steadily add the chicken pieces, skin side down, to the skillet. (Do not overcrowd; if necessary, do in batches.) When pieces have been added, cover the skillet, reduce the heat to medium to medium-high, and set a timer for 7 minutes. After 7 minutes, uncover the skillet, turn chicken, and continue to cook, uncovered, for another 7 minutes.
Turn chicken skin side down and cook for about 5 minutes more, turning to ensure that both sides are golden brown.
As the chicken pieces finish cooking (the juices near the bone will run clear), remove them from the skillet and drain on paper towels. Keep warm in 250 degree F oven until all batches are completed. Serve hot, warm or at room temperature with lemon wedges, if desired. Chicken can be refrigerated for several hours. Reheat, uncovered, on a baking sheet at 325 degree F for about 10 minutes.