Preheat the oven to 350 degrees. Lightly grease a baking sheet with softened butter.
Put the 1 tablespoon butter in a small saucepan. Set the pan on the stove and switch on the heat to low. When the butter has melted, turn off the heat.
Lightly flour a work surface. Unroll the pizza dough on top of the flour. Using a pastry brush, brush the melted butter over the surface of the dough.
Put the granulated sugar and cinnamon in a small bowl. Stir with a small spoon until well mixed. Sprinkle the cinnamon-sugar mixture over the dough, leaving a ½-inch border on all sides.
Starting at the long side, roll up the dough into a log. Using your fingertips, pinch the seam together to seal.
Using a knife, cut the log crosswise into 1-inch-thick slices. Put the slices, cut side up, onto the greased baking sheet.
Using pot holders, put the baking sheet in the preheated oven. Bake until golden brown, about 20 minutes. Carefully transfer the baking sheet to a cooling rack. Cool 5 minutes. With a spatula, transfer rolls to the cooling rack and cool five minutes more.
Put the milk and confectioners' sugar in a small bowl. Stir with the spoon until a smooth frosting forms. Stir in the vanilla. Using a butter knife, spread frosting over the tops of the cinnamon rolls.
