Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Make rolls: Unroll crescent roll dough onto a silicone baking mat or lightly greased work surface and separate into 4 rectangles; firmly press diagonal perforations with your fingers to smooth them out.
Mix together brown sugar and cinnamon in a small bowl; sprinkle evenly over rectangles.
Roll each rectangle into a log starting from the longer side; cut each log into ½-inch-thick slices. Arrange cinnamon rolls close to one another on the prepared baking sheet.
Bake in the preheated oven until golden, 10 to 13 minutes. Transfer cinnamon rolls to a plate and let cool for 1 minute.
While cinnamon rolls are baking, make icing: Stir together confectioners' sugar, milk, and vanilla in a small bowl until smooth.
Pour icing over warm rolls.
