To make the dough, place 1/4 cup of the warm water in a small bowl. Stir in the sugar until dissolved, then sprinkle the yeast onto the surface of the water. Let the yeast sit undisturbed for 1 minute, then stir it into the water until blended. Let the yeast "proof" for 10 minutes; it should become bubbly because of the presence of sugar. If it doesn=t bubble, it is outdated and inactive.
Pour the yeast mixture into a large bowl. Mix in the remaining cup of water, the olive oil, and salt. Add the flour and semolina and stir with a wooden spoon until a ball of dough forms. Turn the dough onto a lightly floured surface and knead 10 minutes, flouring the work surface as necessary to prevent the dough from sticking.
Lightly oil a large ceramic or glass bowl. Place the dough inside, then flip it over so the oiled side is on top. Cover with plastic wrap and place in a warm place to rise until double in bulk, about 11 1/2 hours.
Meanwhile make the topping. Heat the oil in a large skillet over medium heat and add the onions. Cook, stirring frequently, until the onions are golden brown all over, about 20 minutes. Mix in the mushrooms and cook until they are brown and juicy, about 5 minutes. Remove the pan from the heat and season the mushrooms with the thyme, salt, and pepper.
Preheat the oven to 450 degrees. If you are using a pizza stone heat it at least 45 minutes before cooking the pizza on it.
Punch down the dough and dump it onto your work surface. Divide it into 3. Lightly flour the work surface, then with a rolling pin, roll one portion of the dough into an 11inch circle. If you are using a pizza stone, then sprinkle some cornmeal onto a pizza peel and place the disk of dough on it; otherwise, lightly grease a baking sheet and place the dough on that.
Spread 1/3 of the sour cream all over the dough. Sprinkle on 1/3 of the fontina cheese, then scatter 1/3 of the mushroom mixture all over the pizza. Bake 8 to 10 minutes, or until the crust is golden brown underneath. Repeat with the remaining ingredients.