In the bowl of a stand mixer, equipped with a dough hook, combine the warm water and sugar. Sprinkle the yeast over the mixture, giving it another gentle stir. Allow the mixture to rest for 5 minutes, during which time the yeast should begin to foam. If you don't see any foam, discard this mixture and start over with fresh yeast.

Add the flour and sea salt to the foamy yeast mixture. Start mixing on low speed until a rough dough begins to form. Gradually increase the mixer’s speed to medium, letting it work the dough for 5 minutes. You should see the dough becoming stretchy and sticky, starting to cling to the sides of the bowl.

Take a large mixing bowl and brush it generously with 2 tablespoons of the olive oil. Transfer the sticky dough into this oiled bowl, using a spatula to help guide it from the mixer bowl. The dough will be too sticky to handle directly. Gently use your fingers to spread any oil collecting on the edges of the bowl over the surface of the dough. Cover the bowl with plastic wrap or a damp cloth and leave it in a warm place to rise until it doubles in size, which should take between 1 to 1½ hours.

Prepare a 9x13-inch baking dish by brushing it with another 2 tablespoons of olive oil. Once the dough has risen, remove the covering and lightly coat your hands with olive oil. Carefully slide your hand beneath the dough's edge, folding it inward. Continue folding the dough inwards around its circumference until it forms a rough ball that can be lifted from the bowl.

Place the dough into the prepared baking dish, ensuring it's coated with oil on all sides. Gently press the dough out towards the edges of the pan. If the dough pulls back, let it rest for a few moments before pressing it out again. Cover the pan with plastic wrap or a damp towel and allow the dough to rise again for about 45 minutes, until it has doubled in size. Preheat your oven to 425°F about 30 minutes into this rise.

Once the dough has risen, remove the covering. Drizzle the remaining 2 tablespoons of olive oil over the dough’s surface. Lightly coat your fingers with oil, then press them into the dough to create indentations across the surface, making sure to press all the way down to the pan. If desired, sprinkle with flaky sea salt and freshly chopped rosemary. Bake in the preheated oven for 20 to 30 minutes, until the focaccia is golden brown.