Preheat oven to 400ºF. Line a large baking sheet with parchment paper or foil.
On a lightly floured countertop, roll each piece of dough into a 6-by-10-inch rectangle. With a floured chef's knife, cut each rectangle into six 10-inch strips. Make 3-inch lengthwise cuts at one end of each strip, 1/4 inch apart, to form broom straws. Twist and pull uncut portion of strips to form a broom handle (each breadstick will be about 15 inches long). Gently transfer broomsticks to prepared baking sheet; place at least 2 inches apart. Separate cut ends.
Brush broomsticks with milk and sprinkle with Parmesan. Bake until golden, 15 to 18 minutes. Serve warm or at room temperature.
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