Butter a large baking dish and evenly spread out the torn bread to cover the bottom of the dish.
In a large bowl, beat to combine the heavy cream, cheese, ham, milk, scallions, salt, pepper and eggs. Pour the egg mixture over the bread and top with the paprika. Cover and refrigerate for at least 1 hour and up to overnight.
Remove the dish from the refrigerator 30 minutes before baking. Preheat the oven to 350 degrees F.
Uncover the dish and bake until puffed, golden and cooked through, about 45 minutes. Let cool 5 minutes, sprinkle with the chives and serve.