Preheat the oven to 375°F (190°C), and coat a 9-inch by 13-inch (23cm x 33cm) baking pan with nonstick spray. Set aside.
In a bowl, mix together the butter, granulated sugar, flour, cinnamon and 1/2 teaspoon vanilla extract.
Lightly flour a work surface and roll out the pizza dough into a 12- by 15-inch (30cm x 38cm) rectangle. Evenly spread the butter mixture over the dough, making sure to reach the edges. Roll the dough, starting from the long side, and cut into 9 pieces. Place in the pan, leaving space for the dough to expand while baking.
Combine the egg and water, and brush the wash over the rolls.
Bake for 40 minutes.
While the cinnamon rolls cool (at least 15 minutes), make the icing by mixing the powdered sugar, milk and remaining vanilla extract. Drizzle over the rolls and serve.