Allow the dough to rest at room temperature for 30 minutes on a floured surface.
Preheat the oven to 350 degrees F. Spray an 8 ½-by-4 ½-inch loaf pan with nonstick cooking spray.
Combine the brown sugar and cinnamon in a small bowl. Set aside.
Roll out the dough to an 8-by-14-inch rectangle. Spread the butter over the dough. Sprinkle the brown sugar mixture over the butter and pat down lightly. With a long side facing you, make 3 vertical cuts in the dough, creating 4 equal strips, each 8 inches long and 3 ½ inches wide. Place one strip of dough into the prepared loaf pan, sugar-side up. Repeat with the remaining strips of dough, stacking them.
Bake until golden brown and puffed, 28 to 30 minutes. Allow to cool for 5 minutes. Invert the pan onto a serving platter and set aside until just cool enough to slice.
To make the glaze, sift the confectioners’ sugar into a medium bowl. Whisk in the milk and vanilla to make a smooth glaze.
Slice the cinnamon roll loaf into pieces and drizzle with the glaze.
