Open both pizza dough tubes and press them together into one ball. Let it rest at room temperature while you prep the dish.
Preheat oven to 400°F (204°C). Brush melted butter across the bottom and halfway up the sides of a 9x13-inch dish. Set aside a little butter for later.
Sprinkle 2 teaspoons of cinnamon sugar over the buttered dish, then add 1 tablespoon of brown sugar on top.
Roll out the dough and stretch it to fit the dish. Press it into the corners and along the edges with your fingers.
Use a pizza cutter to slice once lengthwise, then cut across to make 16 sticks. Gently press them back into shape if needed.
Brush with the remaining melted butter. Sprinkle on the rest of the cinnamon sugar and brown sugar.
Bake on the center rack for 15–17 minutes, until puffed and golden. Use a fork to check that the center is fully cooked.
Let cool slightly. In a bowl, whisk powdered sugar, vanilla, and 2 tablespoons milk. Add more milk if needed for a smooth drizzle.
Drizzle some glaze over warm sticks and serve with the rest of the glaze on the side for dipping.