Proof the dough - Proof the Birrittella's traditional pizza dough according to the package directions for 3 1/2 hours.
Preheat the oven - Preheat the oven to 400°F. Line 2 baking sheets with parchment paper.
Make the cinnamon sugar - In a small bowl, combine the granulated sugar and ground cinnamon. Set aside, reserving about 1 tablespoon of the mixture for topping the twists.
Roll and cut the dough - On a lightly floured surface, roll the pizza dough into a rectangle. Cut the dough into 4 equal squares. Place 2 squares on each prepared baking sheet.
Add the filling - Brush approximately 3 tablespoons of the melted butter over 1 dough square on each baking sheet. Sprinkle with the cinnamon sugar mixture, reserving about 1 tablespoon for topping.
Add cranberries and pistachios - Evenly distribute the chopped dried cranberries and pistachios over the cinnamon sugar. Gently press them into the filling.
Top and cut - Place the remaining 2 unbuttered dough squares over the filled dough squares. Press gently around the edges to seal. Cut each assembled square into 4 pieces, making 8 pieces total.
Twist - Twist each piece several times. Brush the twists with the remaining melted butter and sprinkle with the reserved cinnamon sugar.
Bake the twists - Bake for 10 to 12 minutes, rotating the baking sheets halfway through. Bake until the twists are golden brown. Oven temperatures can vary, so watch them carefully toward the end of the baking time.
Prepare the glaze - In a small bowl, combine the powdered sugar, milk, vanilla extract, and orange zest. Mix until smooth.
Glaze and serve - Allow the twists to cool slightly, then drizzle the orange vanilla glaze over them while they are still slightly warm.