In a medium-sized mixing bowl, whisk together flour, sugar, baking powder, and salt.
Grate the cold butter into the dry ingredients and mix until crumbly.
Pour in the milk and stir until just combined to form a dough.
On a floured surface, roll the dough into a 10x12-inch rectangle. Mix brown sugar, cinnamon, melted butter, and vanilla extract to create the filling. Spread the filling evenly over the dough, leaving a small margin at the edges.
Roll the dough tightly from one long end to the other. Cut into 10 equal pieces and place in a greased 9-inch baking pan.
Preheat the oven to 375°F and bake the rolls for 25-30 minutes or until golden brown.
While rolls bake, beat cream cheese until smooth, then add confectioners' sugar and milk to make the icing. Add more sugar or milk to reach your desired consistency.
Remove rolls from oven, let cool slightly, and drizzle with icing before serving.