Preheat oven to 375°F (190°C). Brown sausages in butter or oil for 3–4 minutes per side, golden but not fully cooked. Set aside.
Combine warm milk, melted butter, sugar, and yeast; let sit 5 minutes. Add egg, flour, and salt, knead 5–7 minutes until smooth, and let rise 30 minutes.
Mix brown sugar, granulated sugar, cinnamon, and softened butter into a paste. Roll dough into a 12×9-inch rectangle and spread filling evenly.
Cut strips about 1½ inches wide, place a sausage on each, and roll tightly. With pre-made dough, just wrap and pinch closed.
Arrange seam-side down on a parchment-lined sheet and bake 18–22 minutes until golden and puffed.
Whisk powdered sugar, milk, and vanilla until smooth and drizzle over warm rolls.
Serve warm while the glaze melts into the cinnamon crust and the sausages stay juicy.