In a large bowl or stand mixer combine 1/2 cup of the warm water with the sugar and yeast. Stir to combine and let rest for 5 minutes. Add the remaining 1/4 cup warm water, olive oil, salt and vinegar.
Begin mixing on medium-low speed and gradually add the flour --knead for a few minutes, or until the dough is smooth and elastic (it should be slightly sticky, but not so sticky that it sticks to your clean fingertip). Mine is usually ready with 1 1/2 cups flour) Cover the bowl with a dry towel and allow to rest in a warm place for 1 hour.
Preheat oven to 475 degrees F with a pizza stone inside or upside down cookie sheet inside.
Gently punch the dough down. Place a large piece of parchment paper on your counter and use your hands (or a rolling pin, but it’s easier with your hands) to spread the dough into a rectangle about 1/3 inch thick.
Mix melted butter and garlic in a small bowl then spread mixture evenly over dough, all the way to the edges. Top with parmesan and mozzarella cheeses.
Transfer the dough (still on the parchment paper) onto the hot pizza stone in the oven. Bake for 8-12 minutes or until golden and bubbly.
Make Ahead Instructions: Make the dough 3-4 hours before you want to bake breadsticks, storing it in the refrigerator to rise (instead of at room temperature).Freezing Instructions: Make the dough. Spray it with non-stick cooking spray and place it inside a freezer safe bag. Try to get as much air as possible out of the bag. Freeze homemade dough for 2-3 months. Allow to thaw overnight in the refrigerator. Remove to the counter for 30 minutes before forming into breadsticks.