Whisk one qwt. cold water, salt, and sugar in large bowl until sugar and salt dissolve. Add chicken and refrigerate for 30 minutes or up to one hour.
Whisk water, flour, cornstarch, pepper, paprika, baking powder, salt, and cayenne in large bowl until smooth. Refrigerate batter while chicken is brining.
Heat oil in large Dutch oven over medium-high heat to 350 degrees. Remove chicken from refrigerator, pour off brine, and pat dry with paper towels.
Rewhisk batter. Transfer half of chicken to batter. One by one, remove chicken from batter (allowing excess to drip back into bowl) and transfer to oil. Fry chicken, adjusting burner as necessary to maintain oil temperature between 300 and 325 degrees, until deep golden brown and white meat registers to 160 degrees, 12-15 minutes.
Drain chicken on wire rack set inside rimmed baking sheet. Bring oil back to 350 degrees and repeat with remaining chicken. Serve.
Can reheat by placing chicken on wire rack set inside a rimmed baking sheet and warming in a pre-heated 450 degree oven for about 10 minutes.