In a medium bowl or measuring cup, stir together the warm water and the sugar, until the sugar dissolves.
Whisk in the yeast and then let stand until it starts to foam, about 5 to 8 minutes. (if the yeast doesn't foam and the yeast granules, float, the yeast is dead. You'll need to discard and begin again with a fresh amount of yeast and water).
Add the room-temperature water and the olive oil to the foaming yeast concoction. Set aside.
In a food processor, combine the semolina and all-purpose flours and the salt.
With the motor running, add the yeast mixture in a steady stream and then pulse until the dough comes together in a rough mass, about 12 seconds (if the dough does not form a ball, sprinkle with 1 to 2 teaspoons of cold water and pulse again until a rough mass forms).
Let the dough rest in the processor bowl for 10 minutes.
Process the dough again for 30 seconds (you may need to hold the top of the food processor with one hand to hold it steady).
The dough should be tacky to the touch, but not sticky.
Transfer the dough to a lightly floured work surface and form it into a smooth ball.
Place the dough in a large oiled bowl, turn the dough to coat with oil, and cover the bowl with plastic wrap.
Let the dough rise in a warm place until doubled in size and looks spongy, about 2 hours.
Turn the dough out onto a lightly floured work surface, gently punch it down, and shape it into a smooth cylinder.
Divide the dough into 2 equal portions.
Shape each portion into a smooth ball, dusting with flour only if the dough becomes sticky.
Cover both balls of dough with a clean kitchen towel and let rest for 15 minutes before proceeding with your pizza recipe.
(You can freeze the balls of dough in gallon-size zipper lock bag, being sure to squeeze as much of the air as possible out of the bag, for up to 2 months. Thaw the frozen dough for 3 to 4 hours at room temp.).