In a large bowl, combine yeast, water, and sugar. Stir and let it sit for 5 minutes.
In a small bowl, combine olive oil, garlic, thyme, and half the salt.
Add half the oil mixture into the yeast mixture. Set the remaining oil mixture aside for later. Add flour to the yeast mixture and mix by hand to combine.
Add dough to a greased bowl and cover with a towel. Let it rest for 1 hour. The dough should be doubled in size.
While the dough is rising, prepare the caramelized onions. Melt butter in a large pot over medium heat. Thinly slice onion and add to the large pot. Season with salt and sugar.
Reduce heat to medium low and cook until onions begin to caramelize. Stir frequently to prevent burning. This will take 20 minutes.
Preheat the oven to 450°F. Pour the extra oil mixture into the bottom of a 9×11-inch baking dish. Make sure to evenly coat the bottom and sides of the baking dish with the oil.
Place the dough into the baking dish and push the dough to the edges with your hands. Use the end of a wooden spoon or finger to make small indents into the dough.
Top the dough with caramelized onions and cheese. Bake for 20 minutes. Can be served warm or be made in advance.