In the bowl of the stand mixer fitted with a dough hook, combine 1/2 cup of the warm water with the sugar and yeast. Let rest for at least 5 minutes, until the yeast starts to activate and becomes foamy.
Add the remaining water, as well as the olive oil, salt, and vinegar
Mix on medium-low speed, gradually adding flour, for about 7 minutes or until it's smooth and elastic. The dough should be slightly sticky, and spring back when you poke at it lightly.
Cover the bowl with a dry towel and rest in a warm place until doubled in size, roughly 1 to 1 1/2 hours.
Gently punch the dough down and place on a lightly floured counter top. Divide the dough into 2 or 3 equal pieces (depending on how thick you want your crust, and how large your pizza).
Roll and stretch your dough to the desired size and thickness. Transfer dough to a piece of parchment paper, and let it rest for about 10-15 minutes.
Preheat your oven to 450 degrees F. If you have a baking steel or pizza stone, place it in the oven to preheat as well.
Brush the pizza dough lightly with olive oil and prick it all over with a fork to prevent bubbles. Slide the pizza (using the parchment paper) onto your preheated pizza stone, or onto your baking tray and bake the dough for 5-6 minutes.
Remove from oven and add pizza sauce and toppings. Return to the oven and bake for 8-14 more minutes or until the curst is golden and the cheese is bubbly and gently browned.