To make the crispy grissini breadsticks served at many Italian restaurants you should use dough made with butter and whole milk. Pizza dough will always make a somewhat chewy breadstick.
Pizza dough is not made with milk; rather, it is made with water, yeast, high gluten flour, salt, sugar, and oil. These ingredients, along with the long mixing or kneading times, will make dough that bends, is chewy, and is very hard when cooked dry.
Adding butter and milk will create the melt-in-your mouth grissini that is crisp, but not hard. One would make grissini dough similar to pizza dough, but would only mix it until smooth. After the dough rises, it is cut and allowed to rise again. At this point, it is brushed with butter, oil, herbs, or cheese before baking at a medium temperature.