Combine warm milk, yeast, sugar, salt, and melted butter in the bowl of a stand mixer. Add egg and flour and knead into a soft, elastic dough.

Transfer dough onto a work surface, knead for 1 minute, and shape into a smooth ball.

Place dough ball into a lightly buttered bowl, cover, and let rise until doubled, approximately 90 minutes.

Divide risen dough into 8 portions, shape each into balls, flatten into discs, and let proof until almost doubled, about 60 minutes.

Prepare filling by combining cooked sausage, cream cheese, green onions, milk, sage, half the beaten egg, salt, pepper, and cayenne into a paste.

Preheat oven to 375°F (190°C).

Press a well into the center of each dough portion, brushing with egg wash and filling evenly with sausage mixture.

Bake in preheated oven until golden brown, 20 to 25 minutes.