Whisk together 1 quart water, sugar, and ¼ cup of the kosher salt in a large bowl until dissolved. Add chicken pieces to brine mixture; cover and refrigerate at least 1 hour or up to 12 hours.

Remove chicken from brine mixture; pat dry with paper towels. Set aside.

Place ½ cup of the cornstarch in a large ziplock plastic bag. Whisk together egg, buttermilk, baking powder, and baking soda in a medium bowl. Whisk together cornmeal, garlic powder, paprika, cayenne, and remaining 1 teaspoon kosher salt and ½ cup cornstarch in a shallow dish.

Working in batches, place a few chicken pieces in plastic bag with cornstarch; seal and shake well to coat. Using tongs, remove chicken from bag, shaking off excess. Dip chicken in egg mixture. Let excess drip off, and dredge in cornmeal mixture, pressing gently to adhere. Place dredged chicken, skin side up, on a wire rack. Cover with aluminum foil or a clean kitchen towel, and let stand 30 minutes.

Pour oil into a large Dutch oven; heat over medium-high until oil reaches 350°F. Meanwhile, preheat oven to 200°F. Working in batches, carefully add a couple of the larger chicken pieces, skin side down, to hot oil in Dutch oven. Cover and fry, moving chicken occasionally to prevent it from sticking together, until it turns golden brown, about 12 minutes. (Adjust heat under Dutch oven as needed to maintain oil temperature between 300°F and 325°F.) Flip chicken; cook until a thermometer inserted in thickest portion of breasts registers 165°F and drumsticks and/or thighs register 175°F, about 8 minutes. Transfer cooked chicken to a clean wire rack, and sprinkle with sea salt. Transfer to a rimmed baking sheet, and place in oven to keep warm.

Repeat Step 5 with smaller chicken pieces, cooking until meat reaches temperatures as specified, about 8 minutes per side. Remove all cooked chicken from oven, and serve.