In a small bowl, combine warm milk, granulated sugar, and active dry yeast. Allow it to sit for 5-10 minutes until frothy.
In a large mixing bowl, whisk together all-purpose flour and salt. Add melted butter, large egg, and the yeast mixture. Stir until a dough forms.
Transfer the dough to a floured surface and knead for about 5 minutes. Place in a greased bowl, cover, and let rise for 1 hour.
In a bowl, mix cold butter and a little flour until paste-like. Shape into a rectangle on parchment, then chill.
Roll dough into a 12x18 inch rectangle. Place chilled butter rectangle in center, folding dough to seal.
Roll out sealed dough, fold into thirds, and refrigerate for 30 minutes. Repeat twice for flakiness.
In a bowl, mix brown sugar and ground cinnamon for the filling.
Roll dough into a 12x24 inch rectangle. Spread softened butter over surface and sprinkle with cinnamon mixture. Roll tightly, sealing edges.
Cut rolled dough into 12 pieces and place in a greased baking dish.
Cover and let rolls rise for another 30 minutes while preheating oven to 375°F (190°C).
Bake for 20-25 minutes until golden brown.
Whisk together powdered sugar, milk, and vanilla extract for glaze. Drizzle over rolls once slightly cooled.