In a large bowl or stand mixer, add warm water, sugar, and yeast. Stir together with a spoon, then set a timer for 5 minutes.
If your mixture is bubbly and a bit foamy after 5 minutes, congratulations! You didn't kill your yeast. If it's not reacting, then start over.
Add 3/4 teaspoon salt, 1/4 cup olive oil, and 2 and 1/4 cups flour. Stir together with a spoon until a soft dough forms. Then if you are using a stand mixer, switch to the dough hook and knead for 8-10 minutes. If you're kneading by hand, turn onto a well-floured work surface and knead for 8-10 minutes. Add more flour as needed. I used an extra 1/4 cup. You want the dough to be workable but it is still pretty sticky. (See photos)
Grease a large bowl. Scrape the dough into the bowl, form a ball, and turn to coat the top with oil. Cover with plastic wrap and let rise for 45 minutes. (Hang onto the plastic wrap for the second rise.)
Coat a 10-inch cast iron skillet** with 1 tablespoon olive oil. Punch down the dough and press it into the bottom of the pan. Use a sharp knife to score the bread on top in a crisscross pattern.
Brush the top with 1 tablespoon olive oil. Then sprinkle with garlic, rosemary, and 1/2 teaspoon salt.
Cover loosely with plastic wrap and let rise another 20 minutes.
Preheat your oven to 400 degrees F.
When the bread has risen, bake at 400 for 20-25 minutes.
Remove from the oven when the top is light golden brown all over.
Remove the bread from the skillet right away and let cool on a wire rack. If you leave it in the pan the crust will get soggy!
Drizzle with more olive oil, slice into wedges, and serve hot!