Melt the butter in a large broiler-proof skillet over medium-high heat. Add the red onion and cook, stirring occasionally, until softened, about 3 minutes. Add the ham and season with 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the onion starts browning, about 2 minutes. Add the broth and bring to a simmer; cook until the onion softens, about 3 minutes.
Reduce the heat to medium low and stir in the pierogi and broccoli. Cover and cook until the pierogi are warm and the broccoli is tender, about 15 minutes. Meanwhile, combine the sour cream, chives and paprika in a small bowl; set aside. Preheat the broiler.
Uncover the skillet and sprinkle with the cheese. Transfer to the broiler and cook until the cheese is melted and the pierogi are slightly golden, 2 to 3 minutes. Serve with the sour cream sauce.