Activate the Yeast: In a large mixing bowl or the bowl of a stand mixer, combine 1/2 cup of the warm water with the sugar and yeast. Stir gently and allow it to sit for about 5 minutes until the mixture becomes frothy, indicating the yeast is active.
Mix the Dough Ingredients: Add the remaining 1/4 cup warm water, olive oil, salt, and vinegar to the yeast mixture. Begin mixing on low speed, gradually incorporating 1 1/2 cups of flour. Continue kneading for several minutes, either with a stand mixer dough hook or by hand, until the dough feels smooth and elastic but slightly tacky. Add more flour, a little at a time, if the dough is too sticky, up to 2 cups total.
First Rise: Cover the bowl loosely with a clean kitchen towel and place it in a warm, draft-free spot. Let the dough rise for about 1 hour, or until it has doubled in size.
Preheat Your Oven: Set your oven to 475°F (245°C). Place a pizza stone or an inverted baking sheet on the middle rack while the oven heats to ensure a hot surface for baking.
Shape the Dough: After the dough has risen, gently punch it down to release air. Transfer it onto a large sheet of parchment paper. Using your hands (or a rolling pin if preferred), press and stretch the dough into a roughly rectangular shape about 1/3 inch thick.
Prepare the Garlic Butter: In a small bowl, mix together the melted butter and minced garlic until combined.
Add Toppings: Brush the garlic butter evenly over the entire surface of the dough, reaching all the edges. Sprinkle the grated Parmesan cheese first, followed by the shredded mozzarella, distributing them evenly.
Bake the Breadsticks: Using the parchment paper as a sling, carefully transfer the dough onto the preheated pizza stone or baking sheet in the oven. Bake for 8 to 12 minutes, or until the crust is golden brown and the cheese is melted and bubbly.
Serve Warm: Remove from the oven and let cool slightly. Slice into sticks and serve immediately with your favorite dipping sauces such as marinara or ranch.