In the bowl of a stand mixer fitted with the dough hook attachment, combine the warm water, yeast, and sugar. Stir and allow it to sit and bloom (foam and bubble), about 10 minutes. Meanwhile, in a mixing bowl, stir together the flour and salt.
Add 1/4 cup of the olive oil to the yeast mixture, then turn the mixer on low and gradually begin adding the flour/salt mixture. When all the flour and salt have been added, turn the mixer up to medium-high and knead the dough for 5 minutes until elastic. If the dough is somewhat sticky, add flour by the tablespoon until it just pulls away from the side of the bowl.
Oil a mixing bowl generously with extra virgin olive oil and transfer the dough to the bowl. Cover the bowl with a kitchen towel and set in a warm place to rise for 1 hour.
Line a jelly roll pan/large sheet pan with parchment paper. Generously oil the bottom of the pan with extra virgin olive oil. Place the dough gently onto the pan and press it out to fit the size of the pan. Make finger indentations all over the top of the dough by pressing firmly into the dough with the tips of your fingers. Return the dough to a warm place and allow to rise another 45 minutes.
Meanwhile preheat the oven to 425 degrees. When the focaccia is done rising, drizzle the remaining 1/4 cup of olive oil on top of the bread and spread it with your fingers to coat. Sprinkle on the rosemary and sea salt. Bake 20-25 minutes until the bread is golden brown and fragrant. If using the garlic, rub the top of the hot bread with the garlic halves when you just remove it from the oven.
When bread has cooled enough to touch, slice it into cubes and serve it warm with olive oil and balsamic for dipping.