Season chicken with coarse salt and pepper, then rub with garlic. Let sit 15 minutes. In a large zip-top bag, combine flour, paprika, cayenne, and fine salt. In batches, add chicken and shake well until evenly coated. If you have time, let chicken sit 15 minutes.
In a 10-inch heavy skillet, heat oil over medium until it reaches 365 degrees. The oil is ready when a little flour dropped in bubbles and sizzles steadily.
Shake excess flour off a piece of chicken, then carefully place in oil, skin side down. Repeat, adding just a few more pieces to skillet (do not crowd pan). Adjust heat to keep a steady sizzle and maintain 365 degrees. Cover and cook until chicken is browned on bottom, about 7 minutes. Carefully flip each piece, then cover and cook until chicken is cooked through, about 8 minutes.
Drain chicken on paper towels. Repeat with remaining chicken, replenishing and reheating oil between batches, if necessary.