Dissolve sugar in the warm water in the bowl of a mixer fitted with a dough hook. Add yeast and stir, then let sit 5 to 10 minutes, until the mixture is foamy. If not foamy or bubbling, proof a new packet of yeast.
Gradually add the flour, 1/4 cup of olive oil, and salt to the yeast mixture and mix on low. Once combined, knead with the dough hook on medium-low speed for 5 to 6 minutes until dough is soft but sticky. If dough is too sticky to pull away from the sides of the bowl, sprinkle in a little more flour.
Transfer the dough to a clean, lightly oiled surface. Knead by hand for another 2 to 3 minutes until the dough is elastic and can come together into a smooth ball.
Coat the inside of a large bowl with olive oil. Put the dough in the bowl, then flip it to coat both sides with oil. Cover the bowl with plastic wrap and let rise in a warm place until the dough has doubled in size, 1 to 1 1/2 hours.
Gently punch down the dough. Cover with lightly oiled plastic wrap and let rest for 20 minutes.
While dough is resting, heat 1 tablespoon of olive oil in a small skillet. Add the onions, cover and cook over medium heat, stirring occasionally, for 10 minutes. Add the brown sugar, cover and cook, stirring occasionally, until browned. Set caramelized onions aside.
Preheat the oven to 425°F. Set the oven rack in the middle position.
Coat a 12" x 16" rimmed baking sheet with 1/4 cup olive oil. Tip the dough onto the baking sheet and begin pressing it out with your hands to fit the pan, pausing periodically for 3-minute stretches to let the glutens relax. Turn the dough over to coat both sides with oil. When dough has been stretched out to fit the baking sheet, let rest uncovered until puffed up, about 20 to 30 minutes. Just before baking, use your fingertips to deeply dent the dough all over, pressing all the way down.
Cut the tomatoes into rounds and scatter on top with the caramelized onions. Sprinkle the top of the focaccia with the flaked sea salt, rosemary, and thyme and lightly drizzle olive oil on top. Bake until golden brown, about 20 minutes. Let focaccia cool in the pan on a wire rack for about 15 minutes, then transfer to a cutting board and cut into squares. Drizzle a little more olive oil on top before serving (optional).