In a large heat-proof bowl, stir in tapioca flour and Parmesan cheese together until combined; then set aside.
In a small saucepan, stir in together the milk, oil, salt, and chopped rosemary and bring to a boil. Remove from heat and pour the hot liquid mixture over the flour mixture, stirring well with a wood spoon until combined. Mixture will be lumpy. Add eggs on one side of the bowl, break them with a fork and give a quick stir. Then, using your hands, mix them with the lumpy flour mixture until obtaining homogeneous dough, about 3-4 minutes.
Preheat oven to 350 degrees F (about 180 degrees C). Line two to three baking sheets with parchment paper and set aside.
Lightly grease your hands with a drop of vegetable oil. Using a tablespoon as measurement, spoon 1-2 tablespoons of the dough onto your hands, mold into balls and then roll into sticks, 2-inch (mini breadstick) to 4-inch long each (medium breadstick). Arrange them on the lined baking sheets, leaving a space (1 to 2-inches) between them. Bake for about 13-15 minutes, or until tapioca breadsticks are puffy and lightly golden on top. Yield will vary according to the size of the tapioca breadsticks (22-45 breadsticks). Mine were approximately 4 inches long a piece, yielding about 22 breadsticks.
Serve while still warm accompanied by coffee, tea, or any other beverage of your choice. If desired, you can brush a bit of melted butter on top of the tapioca breadsticks. Enjoy!