Pour the warm water in a large mixing bowl. Add the sugar and package of yeast. Stir the mixture slowly until yeast and sugar are dissolved. Let sit to allow the mixture to "mature" for about ten minutes or so. The mixture will begin to react; clouding and forming a foamy "head" on the surface of the mixture.
Add the salt and olive oil and stir again to combine and dissolve the ingredients. Add 1 cup of flour and whisk in until dissolved. Add the second cup of flour and whisk it in. Add the 3rd cup of flour and combine. By now the dough mixture should be fairly thick. Add the last cup of flour and, with your hands, begin to combine and knead the dough.
Remove the dough ball to a tabletop to knead it. You may need to add a dusting of flour from time to time to reduce the stickiness of the dough. Be patient, folding the dough ball in half and then quarters, over and over again for about 8 to 10 minutes, (or about 100 "cycles".) Kneading by hand is laborious, but very important. Better to over-knead than under-knead. You'll know you've done well when the ball no longer sticks to your hands. It will become a smoothly-textured ball slightly larger than a large grapefruit.
Coat the dough ball with a thin layer of olive oil, and place it in the bottom of a large mixing bowl which has also been coated on the inside with olive oil. Stretch a piece of kitchen film over the top of the bowl and set it in a warm place such an as un-lit oven, (ambient temperature of 70°F to 80°F). Allow the dough to rise, undisturbed, for 60 to 75 minutes. The dough will have grown to at least twice its original size.
Take the dough out of the bowl and cut in half with a knife. You now have two pizza dough balls, enough to make two (2) 12" deluxe pizzas. Take each raw dough portion and hand-mold them into balls. Press each doughball flat to squeeze and release any air trapped inside. Form the portions back into balls, smoothing the outer surface and tucking each ball "into itself" from underneath, (like folding a sock into itself), before storing or going on with the next step.
If you wish to store the dough, by either freezing or refrigeration, you can place the dough balls in zip-lock bags. Squirt a little olive oil into each of the bags to keep the balls moist and pliable and to ease removal when ready for use. If you choose to freeze or refrigerate: the dough balls may continue to rise until they are substantially cooled down or frozen, which is OK as long as they don't break out of their bags. If they do, mold them back down into balls and re-bag them.
If you choose to continue with making the pizzas now, here's how. Some dough experts like to "proof" their dough balls at this point. They can be set in a bowl or plastic tray, covered, to "rest" for an additional 15 or 20 minutes if you wish. Some recipes call for up to an additional hour of rising. For practical purposes, pizza dough does not have to be put through a complete second rise cycle.
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