In a small bowl, stir yeast and sugar into water and let mixture stand until bubbling, about 10 minutes.
In bowl of a standing mixer fitted with bread hook, combine flour, 1 tablespoon fine sea salt, ½ cup olive oil, and yeast mixture. Mix at low speed until combined. Once dough pulls away from sides of bowl and comes together, increase speed to medium and continue to knead dough until soft and smooth, 5–6 minutes. Add more flour, as needed, if dough is too sticky. Place dough on a lightly floured surface and knead by hand for another 30 seconds, adding more flour, as needed.
Coat inside of a large bowl with olive oil and place dough in bowl. Cover bowl with a damp cloth and allow to rise in a warm place until doubled in size, about 1 hour.
Coat a 9-by-13-inch baking pan with remaining ½ cup olive oil. (It will seem like a lot of oil.) Place dough on pan and press and stretch dough out to fit pan. Flip dough over to coat other side with olive oil, stretching and pressing to fit pan. As you shape the dough, use your fingers to press holes all the way through the dough, creating the craggy surface characteristic of focaccia. Set pan in a warm place until dough has doubled in size again, about 1 hour. Near the end of the rising time, heat oven to 425°.
Liberally sprinkle top of dough with coarse or flaky sea salt and rosemary, and lightly drizzle olive oil on top. Bake until top is a golden brown, 25–30 minutes. Let focaccia cool on a wire rack before cutting and serving.