In a mixing bowl, combine the sourdough starter, milk, and sugar.
Add the softened butter and egg, mixing until well combined.
Gradually incorporate the flour and salt, kneading until a soft dough forms.
Cover the dough and let it rise in a warm place for about 1 hour.
Roll the dough out on a floured surface into a rectangle (about 1/4 inch thick).
Spread softened butter over the dough, followed by a mixture of brown sugar and cinnamon.
Tightly roll the dough into a cylinder, sealing the edges.
Cut the roll into 12 equal pieces and place them in a greased baking dish.
Cover the dish and let the rolls rise for another 30 minutes.
Bake in a preheated oven at 350°F (175°C) for 25-30 minutes until golden brown.
Remove from the oven and drizzle with icing made from powdered sugar and a splash of milk.