Cinnamon-Roll Cake

Ingredients

• Cinnamon Roll
• 8 tablespoons unsalted butter, at room temperature, divided
• 1 cup brown sugar
• ¼ cup all-purpose flour
• 1 tablespoon ground cinnamon
• ½ teaspoon pure vanilla extract
• 2 pounds pizza dough (store-bought or homemade )
• 1 egg
• 1 tablespoon water
• Icing
• 1 cup confectioners' sugar
• 3 tablespoons whole milk, or more as needed