Begin by melting the butter in a large skillet over medium-high heat. Add the chicken pieces and cook until each side is lightly browned, approximately 3 to 5 minutes per side. Once browned, incorporate the chopped onion, cooking until it becomes translucent, which should take around 5 to 8 minutes.

Pour in the water, and season with paprika, ground black pepper, and salt. Continue to cook until the chicken is thoroughly cooked—no longer pink at the bone, and juices run clear—about 10 to 15 more minutes. The internal temperature near the bone should reach at least 165°F (74°C). Remove the chicken from the skillet and keep it warm.

While the chicken cooks, bring a large pot of water to a rolling boil over high heat. In a large bowl, combine the water, beaten eggs, and salt. Gradually mix in the flour to create a stiff batter. Using two spoons, scoop and scrape small portions of batter into the boiling water. Repeat until several dumplings are cooking simultaneously. Cook until the dumplings rise to the surface, about 10 minutes. Remove them with a slotted spoon and drain in a colander, rinsing with warm water and draining again. Repeat this process for the remaining batter.

In a small bowl, mix the sour cream and flour until smooth. Gradually add this mixture to the skillet, stirring slowly to integrate it with the sauce. Bring the sauce to a boil, continuing to stir constantly until it thickens, which should take about 2 to 3 minutes.

Gently fold the dumplings into the sauce, ensuring they are well coated. Heat everything through for another 2 to 3 minutes. Serve the dumplings, sauce, and chicken on plates, ensuring each serving has an ample amount of both sauce and chicken. Enjoy your homemade Chicken Paprikash!