For the chicken:
½ cup water
½ cup apple cider vinegar
¼ cup canola oil, more for the grill
3 tablespoons hot sauce
1 tablespoon Worcestershire sauce
1 tablespoon coarse salt, plus more for seasoning the chicken
1 teaspoon garlic powder
1 (4- to 5-pound) chicken, cut into 8 pieces
Freshly ground black pepper
For the burgers:
3 pounds ground beef
Coarse salt and freshly ground black pepper
4 ounces crumbled bleu cheese
8 buns, toasted, for serving
Sliced tomatoes, romaine lettuce, very thinly sliced Vidalia onion, for serving
Combine the water, vinegar, oil, hot sauce, Worcestershire sauce, salt and garlic powder in a large bowl. Remove about ¼ cup of the marinade and set aside. Season the chicken with salt and pepper. Add the chicken to the remaining marinade and toss to coat. Refrigerate while you prepare the burgers and the grill.
Place the ground beef in a bowl and season with salt and pepper. Divide the mixture into 8 equal patties. Make a depression in the center of each patty and place 1 tablespoon cheese in the center of the meat. Form meat around the cheese to enclose it. Then, gently press down on the center creating a slight depression in each burger; repeat with remaining portions of meat. This will prevent the burger from having a bulge and the burger will be thick but flat and evenly cooked. Place in the refrigerator while you prepare the fire.
Prepare a charcoal fire using about 6 pounds of charcoal and burn until the coals are completely covered with a thin coating of light gray ash, 20–30 minutes. Spread the coals evenly over the grill bottom, position the grill rack above the coals and heat until medium-hot (when you can hold your hand 5 inches above the grill surface for no longer than 3 or 4 seconds). Or, for a gas grill, turn on all burners to high; close the lid and heat until very hot, 10–15 minutes.
Remove the meats from the refrigerator to take the chill off. Scrape the hot grill rack clean with wire brush. Lightly dip wadded paper towels in the oil; holding the wad with tongs, wipe the grill rack.
Remove the chicken from the marinade; discard marinade. Use the reserved marinade to brush the chicken while cooking. Place the chicken on the grill and cook until seared, about 1–2 minutes per side for legs and thighs, and 3 or so minutes for breasts. Move the chicken to medium-low heat or reduce the heat to medium; continue to grill, turning occasionally and brushing with the reserved marinade, until the juices run clear when pierced, 12–18 minutes.
Grill patties, uncovered, without pressing down on them, until well seared on the first side, about 3 minutes. Flip burgers with a barbecue spatula; continue grilling until medium rare, an additional 3–4 minutes. Transfer burgers to a warm platter.