Make the dough: Combine the flour and salt in a large bowl. Make a well in the center and pour in the warm water. Sprinkle the sugar and yeast over the water and let stand until bubbling, 5 to 10 minutes. Pour the olive oil into the well, then stir with a wooden spoon to make a shaggy dough. Turn out onto a lightly floured surface and knead until smooth and elastic, about 1 minute. Place in a lightly oiled bowl, tightly cover with plastic wrap and let rise in a warm place until doubled in size, about 2 hours.
One hour before baking, put a pizza stone or large inverted baking sheet on the lowest oven rack and preheat to 500 degrees F. Make the sauce: Heat the olive oil in a medium saucepan over medium heat. Add the garlic and red pepper flakes and cook, stirring, until slightly softened, about 1 minute. Add the tomato puree and 2 cups water. Stir in the oregano and 1 teaspoon salt. Bring to a boil, then reduce the heat and simmer, stirring occasionally, until thickened, about 20 minutes. (The sauce will keep in an airtight container in the refrigerator for up to 1 week or in the freezer for up to 2 months.)
Divide the pizza dough into 2 balls. On a lightly floured surface, stretch 1 ball of dough into a 12- to 14-inch round. Transfer the dough to a piece of parchment paper, then slide the parchment with the dough onto a pizza peel or inverted baking sheet.
Spread about 1/2 cup of the sauce on the crust in a thin layer, then scatter 1 cup mozzarella on top. Slide the pizza (on the parchment) onto the hot stone. Bake until the crust is golden brown and the cheese is bubbling, 8 to 10 minutes. Let cool 1 to 2 minutes before slicing. Repeat to make the second pizza.
Low-moisture mozzarella (either part-skim or whole milk) is best for covering this pie. It's saltier than fresh mozzarella and it's extra stretchy.