Prepare the Dough: In a large mixing bowl, combine all-purpose flour, granulated sugar, active dry yeast, and salt, mixing well. Gradually add warm milk, room temperature butter, and a large egg, stirring until the mixture is smooth and cohesive. Knead the dough for about 5-7 minutes until it becomes soft and elastic. Cover the bowl with a clean cloth and let it rise in a warm place for about 1 hour or until doubled in size.

Make the Filling: While the dough is rising, crush the chocolate sandwich cookies into coarse crumbs in a separate bowl. Mix these crumbs with brown sugar, softened butter, and a splash of vanilla extract until you have a spreadable filling. Ensure the mixture is well combined, creating a rich, cookie-flavored paste.

Roll Out the Dough: Once your dough has risen, punch it down gently to release the air. On a lightly floured surface, roll out the dough into a rectangle about ¼ inch thick. Spread the cookie filling evenly over the surface, leaving a small border around the edges.

Assemble the Rolls: Starting from the longer edge, carefully roll the dough tightly into a log shape. Pinch the seam to seal it. Slice the log into even pieces, about 1-2 inches thick. Place the slices in a greased baking dish.

Let the Rolls Rise: Cover the baking dish with a clean kitchen towel and let the sliced rolls rise for an additional 30 minutes.

Bake the Rolls: Preheat your oven to 350°F (175°C). Bake for approximately 20-25 minutes, or until golden brown.

Prepare the Icing: While the rolls are cooling, mix powdered sugar with a splash of milk in a small bowl. Whisk until the icing reaches a smooth, drizzle-worthy consistency.

Glaze and Serve: After allowing the rolls to cool slightly, drizzle the icing generously over the warm rolls.