Heat the 3 quarts (12 cups) vegetable oil in a large Dutch oven or other deep-sided pot over medium-high heat until it reaches 350°F (176°C). Place a wire rack on a rimmed baking sheet.
Discard the brine and pat the chicken pieces completely dry with paper towels.
Whisk the batter to recombine. (If the batter seems too thick, add some cold water, no more than 1 tablespoon at a time, until the batter becomes the consistency of pancake batter.)
Place half the chicken pieces in the batter and turn to coat. Remove the chicken from the batter, allowing the excess to drip back into the bowl, and carefully slip it into the oil.
Fry the chicken and keep your attention on the oil temperature, adjusting the heat as needed to keep the oil between 300°F and 325°F (149°C and 163°C).
Cook the chicken until deep golden brown and the white meat registers 160°F (71°C) or 175°F (79°C) for dark meat, 12 to 25 minutes, depending on the size of the chicken pieces and the temperature of your oil.