Place water, sugar, and yeast in the large mixing bowl of a stand mixer. You can give it a quick stir or two, just to get the yeast wet. Then leave it and let it sit til foamy on top (5-10 minutes).
Turn the mixer on low speed and add 1 cup of flour and the salt. Mix until combined. Add oil and mix well.
Finally, gradually add as much of the remaining cup of flour as you can (it's okay if it's only like 3/4 of it) and mix until the dough pulls away from the sides of the bowl.
Preheat the oven to 220 F and then turn it off (keep the door closed) and grease the skillet.
Place dough onto a floured surface and knead the dough few times until you feel it become smooth and not sticky anymore.
Shape into a ball and then roll out the dough into a circle the size of your skillet. Place in skillet and stretch up the sides. Cover with a kitchen towel and place in the (turned-off) oven for 20 minutes to rise.
Take the skillet out of the oven once the dough has risen and preheat the oven to 400 F.
While the oven is heating, make indentations in the dough with your fingers. Mix the melted butter, parmesan and seasoning in a small bowl.
Brush the dough with all of the butter-herb-parmesan mixture, then sprinkle some sea salt on top (just a few pinches).
Bake for around 20 minutes, until golden brown. Once done, remove from the oven and brush with the additional melted butter or olive oil. Let cool until safe to touch and slice.