Add water to a large bowl or the bowl of a standing electric mixer. Scatter yeast over the water’s surface. Stir to combine. Let stand until bubbly, about 5 minutes.
Add two cups of the flour, the olive oil, salt, and sweetener to the bowl and mix with a wooden spoon or the mixer's paddle attachment until well combined. Add remaining flour, about ½ cup at a time, mixing by hand or with the paddle until the dough clears the sides of the bowl. If working by hand, turn the dough out on a lightly floured surface and knead, adding flour a little at a time if the dough still feels sticky. Knead until dough is smooth and elastic. If using an electric mixer, switch to the dough hook and knead on medium speed for about 5 minutes until the dough is smooth and elastic. If the dough feels too sticky, add a bit more of the flour and knead for another minute or two. Transfer dough to a bowl coated with olive oil, turning the dough once to coat it with oil. Cover with plastic wrap and let it rise in a warm place until doubled in volume, about an hour.
Preheat oven to 475º F. Grease and dust two baking sheets with cornmeal or place a baking stone in the oven (preheat for at least 45 minutes before baking). Gently deflate the dough and turn it out on a lightly floured surface. Divide the dough in half (or into as many pieces as you'd like pizzas) and shape each piece into a ball. Cover with plastic wrap and let the dough rest for 10 to 15 minutes as you prepare the toppings.
Working with one at a time, flatten each dough ball on a lightly floured surface, rolling and stretching it into a 12-inch round for large pizzas. Place the dough circles on prepared baking sheets or pizza paddles dusted with cornmeal.
Brush the top of each dough round generously with olive oil. Top with sauce, toppings, and cheese as desired.
If using a baking stone, working one at a time, slide pizzas off the paddle (a baking sheet works just as well if you don't have a paddle) onto the baking stone. If using baking sheets, place the pans and pizzas in the oven to bake. Bake until the crust is browned and the cheese is melted and golden around the edges, about 12 minutes. Cool pizzas for 10 minutes before slicing.