Heat oven to 375°F. Grease an 8- or 9-inch square or round baking pan.
Stir brown sugar and cinnamon together for the filling. Set aside the melted filling butter separately.
Whisk flour, baking powder, granulated sugar, and salt in a large bowl.
Combine milk, melted butter for the dough, and vanilla. Pour into dry ingredients and stir until a soft dough forms.
Turn dough onto a lightly floured surface. Knead gently 5–6 times until smooth.
Roll dough into a 10x12-inch rectangle, about 1/4-inch thick.
Brush dough with the melted filling butter. Sprinkle evenly with the cinnamon-sugar mixture.
Roll up tightly from a long side to form a log. Pinch seam to seal and trim uneven ends if needed.
Slice into 9 rolls. Arrange in the pan with slight spacing.
Bake 20–24 minutes until puffed and lightly golden at the edges. Cool 5 minutes.
Whisk powdered sugar, milk, vanilla, and a pinch of salt until smooth. Adjust milk for desired drizzle.
Drizzle glaze over warm rolls. Serve right away.