Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment.
Unroll one tube of dough onto one of the prepared baking sheets into a rectangle. Repeat with the other dough and other baking sheet. Refrigerate both while you make the filling.
Combine the butter, brown sugar and cinnamon in a small bowl. Mix together with your hands or a rubber spatula until completely combined.
Spread the filling on one of the dough rectangles, leaving a 1-inch border. Pick up the other dough rectangle, keeping it on the parchment (it will stick to the parchment, making it easier to transfer). Arrange it parchment-side up on top of the other dough piece to sandwich them together, lining them up as evenly as possible. Peel away the top piece of parchment. Brush the dough with the beaten egg and use a fork to crimp the edges.
Bake until golden brown and crispy, 15 to 20 minutes. Let cool for 5 minutes before icing.
Put the confectioners' sugar in a bowl. Stir the milk and vanilla together in a separate bowl, then slowly whisk the milk mixture into the sugar until thick but pourable.
Drizzle the icing over the top of the pastry in a zigzag pattern. Cut into squares and serve warm.
This recipe uses crescent dough sheets, but you can also use crescent roll dough. When you unroll the tubes of dough, just press together the perforated lines on the sheets so there are no holes.
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