Make the pickled radish cubes: Combine the first four ingredients in a large bowl, whisking until the sugar and salt are dissolved. Add the radish and toss to coat. Leave at room temperature, covered, for 24 hours. Then refrigerate.
Make the pre-coating: In a large bowl, whisk together the first 4 ingredients. Add the chicken and toss well until evenly coated in all areas. Transfer the chicken to a rack, shaking the chicken well to get rid of any excess coating. Let rest uncovered for 1 hour.
Pour enough oil into a 6-quart Dutch oven to reach a depth of 2 inches. Heat the oil over medium-high heat until a deep-fry thermometer reads 350 degrees Fahrenheit (175 degrees Celsius).
Make the batter: In a large bowl, whisk together the dry ingredients. In a smaller bowl, whisk together the wet ingredients. Just before frying, whisk the wet mixture into the dry mixture. The consistency should be relatively thin and runny. Working in two batches, dip each piece of chicken into the batter, letting excess batter drip off. Suspend the chicken in the oil for a couple of seconds for it to set before letting it slip completely into the oil, otherwise the chicken will fall and stick to the bottom of the pot. Fry the chicken until golden brown and cooked through, 15 to 20 minutes per batch. Transfer to a wire rack to drain as done.
Make the BBQ sauce: Combine all of the ingredients in a small saucepan and simmer over medium heat until slightly thickened, 3 to 5 minutes. Serve the chicken with the BBQ sauce, pickled radish and beer.
The batter's dry and wet ingredients can be assembled but not mixed together ahead of time. Whisk the wet and dry ingredients right before you¿re ready to fry the chicken, otherwise the batter may thicken too much. Cook the chicken in batches, cooking the drumsticks and thighs together as one batch and the wings as a separate batch. Loosen the BBQ sauce with a little water if you want to drizzle the sauce or glaze the chicken.