For the dough and filling: Lightly butter two 8-inch round baking pans.
On a lightly floured surface, roll out the dough to a 15- by 7-inch rectangle. In a small bowl, combine the brown sugar and cinnamon; set aside. Spread the softened butter over the dough. Sprinkle the dough evenly with the cinnamon-sugar mixture. Starting at the long edge, roll up the dough as for a jelly roll. Pinch the seam to seal. Cut the rolled dough into 16 slices and divide the slices between the prepared baking pans. Set aside in a warm place and allow the dough to rise until doubled in size, about 1 hour.
Preheat the oven to 400 degrees F. Bake until golden on top, about 15 minutes. Promptly invert the baking pan over a wire rack and lift the pan from the buns. Scoop any filling that has remained in the pans over the buns. Allow the buns to cool slightly, about 10 minutes.
Meanwhile, for the icing: In a medium bowl, beat the butter, sugar, cream cheese and vanilla with an electric mixer until fluffy. Spread the icing over the rolls. Serve warm.