Arrange the chicken in a baking dish or tray. Whisk the buttermilk, 1 teaspoon salt, and the hot sauce together and pour over the chicken. Cover, refrigerate, and marinate for at least 3 hours or up to 8 hours.
When you're ready to fry, whisk together the flour, 1 tablespoon salt, and the black pepper and pile onto a large plate or tray. Arrange the chicken and the flour next to the stovetop.
Add about ½ inch oil to a large heavy skillet at least 12 inches in diameter. If your pan is smaller, use two pans or fry the chicken in batches. Heat the oil over medium-high heat to about 375°F. (If you don't have a thermometer, drop a 1-inch piece of bread into the oil. It should take about 60 seconds to become deep golden brown.)
Pick up a piece of chicken, let the buttermilk drip off, then dredge the piece through the seasoned flour, making sure all the creases and corners are well coated. Add each piece to the hot oil carefully, skin side down. Continue with all the chicken, keeping an eye on the temperature. You want the oil to sizzle in a lively fashion, but you don't want the chicken browning too quickly.
Once all the chicken is in the pan, turn the heat to medium so the sizzling is steady but not too wild and then cover the pan. After about 15 minutes, remove the lid and check the chicken. If a rich brown crust has developed, turn each piece. If the chicken still looks pale, increase the heat a tiny bit and cook for a few more minutes before turning. Continue frying the chicken until well browned on the second side and the meat is totally tender when pierced with a knife, 30 to 40 minutes total. Don't let the browned bits in the bottom of the pan burn.
Remove the pan from the heat and transfer the chicken to a rack.
To make the gravy, skim off all the fat from the skillet except about a tablespoon, capturing any crisp bits in the skimmer and reserving them. Put the pan back over medium-high heat, add the flour, and with a flat whisk blend with the fat. Add the broth and whisk until the browned bits in the pan are dissolved. Simmer until reduced by about half, 3 to 4 minutes, whisking frequently. Add the milk and cream, and continue simmering until slightly reduced, creamy, and the flavor is deep and mellow. Add back any reserved crisp bits, squeeze in a few drops of lemon juice to taste, and adjust the salt and pepper. Serve in a gravy pitcher with the chicken pieces.