Whisk together the water and sugar in a small bowl. Sprinkle the yeast overtop and let sit 5 minutes, then stir and let sit another 5 minutes.
In a large mixing bowl or bowl of a stand mixer with a hook attachment, mix together 4 cups of flour and salt. Add in yeast mixture and 1/3 cup of oil and knead about 5 minutes, adding flour as necessary. The dough should be soft but shouldn't stick to your fingers.
Coat a large mixing bowl lightly in oil and place the dough in the bowl, turning it around so that the dough is lightly coated in oil. Cover and let sit about an hour, until doubled in size.
Pour 1/4 cup of oil into a 10"x17" baking sheet. Spread to fill the sheet. Once the dough has doubled, place it on the pan and press the dough to fill the sheet. Cover with plastic wrap or a clean towel (so that it sets directly on top of the dough) and let sit another hour, until it has doubled again.
During the last ten minutes of the second rise, preheat the oven to 425°F.
Once the dough has doubled again, spread remaining 2 tbsp of oil overtop. Use your fingers to press dimples all around the dough. Sprinkle the sea salt and rosemary overtop.
Bake for 22-26 minutes, until crisp to your liking. Eat warm or allow to cool before cutting into.