Season and chill the chicken. Pat the chicken dry with paper towels without rinsing it. Combine the chicken salt, black pepper, and sweet paprika. Rub the pieces with the measured chicken-coating oil, then season all over. Cover and refrigerate for 20 minutes. Keep raw poultry separate from other food and clean hands, tools, and work surfaces afterward.
Prepare the oven and pan. While the chicken chills, preheat a conventional oven to approximately 390°F (200°C), with a rack in the middle. Coat a roughly 14-inch (35–36-cm) round metal baking pan with the separate pan oil. Arrange the chicken with a little space between pieces. For larger scales use additional comparable pans; do not make one deep casserole.
Prebake the chicken. Bake for 20 minutes, turning the pieces once after about 10 minutes to begin coloring both sides. This is only the first cooking stage. Continue directly with the batter and the remaining bake; do not cool partially cooked chicken for finishing later.
Mix and rest the batter. During the chicken’s first baking stage, whisk the eggs with the batter salt. Whisk in about half of the batter milk. Combine flour and baking powder, then whisk them into the egg mixture in portions. Gradually add the remaining batter milk, followed by the sparkling water. The batter should be smooth and pourable, a little thicker than a very thin crêpe batter, not a kneadable dough. Rest it for 5 minutes during the same prebaking interval, then stir briefly.
Pour around the chicken. Carefully remove the hot pan using oven mitts. With tongs, lift the chicken pieces slightly to loosen any that stick. Pour the batter evenly between and partly beneath them, keeping as much of the upper skin exposed as possible. Leave the cooking juices in the pan; they flavor the baked batter.
Bake the chicken and batter together. Increase the conventional oven to approximately 430°F (220°C). Bake for another 25–30 minutes as an initial guide, until the batter is puffed, fully set, and deep golden with some crisp darker edges. Check representative thick chicken pieces away from bone and the center of the batter with a food thermometer: both must reach at least 165°F (74°C). Continue baking if either is under temperature or the batter is wet and raw inside. If everything is cooked but the batter is pale, allow about 3–5 extra minutes while watching closely.
Whisk the cultured-cream finish. While the main bake finishes, whisk the cultured sour cream with the separate finishing milk and its small pinch of salt until smooth and pourable. Keep chilled until ready to use. This is the measured finishing mixture, not extra batter.
Pour and warm the finish. Once the chicken and batter are fully cooked, remove the pan and switch the oven off. Pour the cultured-cream mixture over the hot baked batter without needing to bury all the chicken skin. Return the pan to the switched-off, still-hot oven for about 5 minutes so the finish warms and partly soaks in. Do not boil it away or continue cooking until the dish dries out.
Rest and portion. Remove the pan and rest for 5 minutes. Cut or scoop the batter between the chicken pieces. Divide the cooked meat, batter, and cream finish evenly among the main portions shown, carving larger pieces if needed. Remember that the chicken is bone-in and serve promptly.