Soak the chicken in the ice water for 30 minutes; drain and pat dry. Preheat the oven to 200°. Fill a large pot with enough oil to reach a depth of 2 inches and heat until it registers 365° on a deep-fry thermometer.
Meanwhile, in a large bowl, whisk together the beer and eggs. Whisk in 2 cups flour, the cornstarch and 1/2 teaspoon salt until smooth. Place the remaining 1/2 cup flour in a shallow bowl. Set the beer batter and the flour next to the hot oil.
Season the chicken with salt and pepper, then coat with the flour. Transfer to the beer batter and coat. Working in batches and using tongs, lift each piece and let the excess batter drip off before submerging in the oil. Fry, turning occasionally and maintaining a temperature of 365° at all times, until cooked through, about 15 minutes for the legs, thighs and wings and 10 minutes for the breasts. Using a slotted spoon, transfer to paper towels to drain. Season with salt and transfer to the oven to keep warm.