Dissolve yeast in ½ cup of water (lukewarm) in a medium-sized bowl.
Place half of the flour (sifted) into a large mixing bowl of a stand mixer fixed with the bread hook attachment. Add 1 tbsp. olive oil, 1 tsp. salt, and the water and yeast mixture. Mix on a medium-low speed for 2-3 minutes until well combined.
Gradually, add remaining flour (sifted) and water and continue mixing until the dough is soft but not sticky. You may not need to use all of the flour and/or water. If the dough is too sticky, add some flour. If the dough is too dry, add some water (1 tsp. at a time). Once a manageable dough is formed, continue mixing with the bread hook attachment on medium-low speed for 10 minutes. Then form into a ball.
Use a pastry brush to grease a 10x14 inch (or equivalent) baking dish with some olive oil. Place the kneaded dough ball onto the baking dish. Cover with a damp tea towel and allow to proof for 1½ to 2 hours (up to overnight) in a warm place or until the dough has doubled in size.
Once the dough has doubled in size, stretch the dough out to both ends to cover the whole baking dish. Cover the dish with a damp tea towel and allow to rise for another 30-45 minutes. In the meantime, prepare the topping and preheat the oven to 230°C (350F).
Topping: Thinly slice onions. Melt 2 tbsp. butter in a large skillet on medium heat, place sliced onions, coat well, and sauté for around 30 minutes, until caramelized to a rich brown color. Stir and scrape frequently (every 4-5 minutes) using a wooden spoon. Ten minutes into sautéing, add 2 tsp. brown sugar and mix. You may need to lower down the heat if onions begin to burn. If onions start to dry out, add 1 tbsp. of olive oil.
After rising time has elapsed, poke the focaccia dough using your fingers to create tiny "dimples".
Mix 2 tbsp. olive oil and 1 tsp. water in a small bowl. Using a pastry brush, brush olive oil mixture over focaccia. Oil mixture will pool into the little "dimples".
Top focaccia with sliced brie and caramelized onions and bake for 14 minutes in a preheated oven. Turn the oven temperature down to 220°C (430F), turn baking dish around, and bake for another 5-7 minutes. Transfer out of baking dish, and slice into squares using a pizza cutter. Serve warm.