Attach the dough attachment to your stand mixer. In the bowl, beat together (on LOW) the flour, room-temperature water, and 1½ tsp sugar until a rough, shaggy dough forms, about 3 minutes. Remove the dough attachment, set it aside (you'll be using it again, soon), and cover the bowl with a kitchen towel and let stand while you prepare the yeast mixture.
In a small bowl, add the warm water. Whisk in the yeast and the remaining 1½ tsp of sugar. Let stand until foamy, about 8 - 10 minutes.
Take the towel from the mixer. Re-attach the dough attachment. With the mixer on LOW, add the yeast mixture a little at a time until it's all been added. Stop the mixer, remove the attachment (keep it handy, though), and massage the liquid from the yeast mixture into the dough. Don't worry if it doesn't get 100% incorporated. The dough will be very wet.
Re-attach the dough attachment and sprinkle in the Kosher salt (1 tbsp plus 1½ tsp). Turn the mixer on MEDIUM speed and beat until the dough is elastic, sticky, and pulls away from the sides of the bowl, about 5 minutes.
Add 3 tbsp of olive oil to a large bowl. Swish the oil around to coat the sides. Lightly oil your hands, and scrape the dough from the mixer bowl into the oiled bowl. Loosely cover with plastic wrap and place in a warm, non-drafty area until doubled in size, about 2½ hours.
Add another couple of tablespoons of olive oil to an 18"x13" rimmed baking pan.
Take the plastic off of the dough and gently use your hands to deflate the dough. Scrape the dough onto the oiled pan. Use your fingers to stretch the dough toward the edges of the pan. Most likely, the dough will want to pull back to the center. If so, cover with a couple of pieces of plastic wrap and allow the dough to rest for 10 minutes. Remove the plastic and try again. If the dough doesn't completely reach the corners, this is fine. Cover with plastic wrap and place the pan in the refrigerator for 2 hours, or overnight.
Remove the pan from the refrigerator and place it back in the warm, non-drafty area for 1 hour.
Preheat your oven to 450°F.
Remove the plastic. Dip your fingers in a little bowl of olive oil, and then press your fingertips firmly into the dough to make deep dimples all over the dough.
Sprinkle on the chopped rosemary and sea salt.
Place in the oven. After 10 minutes, drizzle 2 more tablespoons of olive oil all over the surface, and then rotate the pan 180°. Bake for another 10 minutes, or until the top is golden brown.
Place the pan on a baking rack and let cool for at least 10 minutes. Brush the surface of the focaccia with a little more olive oil. Cut into slices and serve warm or at room temperature.