Preheat oven to 350°F. Roll dough out on a parchment paper–lined work surface into a large (14- x 11-inch) rectangle (about 1/8 inch thick), flouring top of dough as needed to prevent sticking.

Process ice cream, milk, chocolate syrup, vanilla bean paste, and salt in a blender on medium speed until smooth, 15 to 20 seconds. Drizzle additional chocolate syrup on the inside of the chilled ...