In the bowl of a mixer gently combine the warm water, sugar, and yeast. Let sit for 5 minutes until foamy.
Add in the 2 teaspoons of salt, flour, and 3 Tablespoons of olive oil. Mix until fully combined, then mix for 5 additional minutes. The dough will be very soft.
Grease a new bowl with a Tablespoon or more of olive oil. Dump the dough into the greased bowl. Cover with a lint free towel and let rise for 1 hour.
While the dough is rising, slice the onions thinly and sauté in some olive oil until they become translucent.
When the onions are translucent, add in the butter. Once melted, add in the sliced mushrooms, black pepper, ¼ teaspoon salt, and 3 sprigs of fresh thyme.
Sauté the mushrooms until they are golden, then add in the marsala wine. Cook until all the liquid has been absorbed by the mushrooms and onions. Remove from heat and set aside.
Coat the bottom of a 9x13 baking dish with olive oil.
Once the dough has finished the first rise, dump the dough into the prepared 9x13" baking dish. Spread the dough out evenly and use your fingers to poke tons of holes in the dough.
Sprinkle the parmesan cheese over the top evenly. Then spread the mushroom and onion mixture over the top of the cheese.
Remove any of the leftover stems from the thyme. Place one fresh sprig of thyme on the top center of the dough.
Cover with a lint free towel and let rise for 30 more minutes.
While the dough is rising, preheat the oven to 400 degrees F.
After the 30 minute rise, bake in the preheated oven for 25 to 30 minutes.