Combine the warm water, yeast and sugar in a small bowl. Put the bowl in a warm place until the yeast is bubbling and aromatic, which takes five to 10 minutes.
In the bowl of a mixer fitted with a dough hook, combine the flour, 1 tablespoon of kosher salt, ½ cup olive oil and the yeast mixture on low speed. Once the dough has come together, continue to knead for five to six minutes on a medium speed until it becomes smooth and soft. (Give it a sprinkle of flour if the dough is sticky and tacky.)
Transfer the dough to a clean, lightly floured surface; then knead it by hand one or two times. Again, give it a sprinkle of flour if the dough is sticky and tacky.
Coat the inside of the mixer bowl lightly with olive oil and return the dough to the bowl. Cover it with plastic wrap and put it in a warm place until the dough has doubled in size, at least one hour.
Coat a half sheet pan with ½ cup olive oil. Put the dough onto the pan and begin pressing it out to fit the size of the pan. Continue to stretch the dough to fit the pan. As you are doing so, spread your fingers out and make finger holes all the way through the dough. Return the dough to the warm place for about 30 minutes.
Preheat the oven to 400 degrees F.
Poke fingers all over the top of the dough all the way to bottom of pan. Arrange prepared veggies and herbs and lightly press to dough. Brush dough and vegetables with olive oil. Bake in preheated oven for 30 to 35 minutes.