For the pizza dough: Combine the water, sugar and yeast in the bowl of a mixer with a dough hook attachment. Let sit for 2 minutes to activate the yeast; it will begin to bubble. On low speed, add the flour and salt slowly, then add the olive oil. Once the dough is incorporated, about 4 minutes, remove from the bowl and knead on a lightly-floured surface for about 5 minutes. Place in a lightly-oiled bowl. Let sit in a warm place covered with a damp cloth until the dough has doubled in size, about 1 hour.
For the cheese sauce: Cook the onions and garlic in the olive oil in a large saute pan until translucent, 2 to 4 minutes. Add the heavy cream and simmer until the cream has reduced by half, 12 to 15 minutes. Turn off the heat and add the cheeses. Stir very well until all the cheese has melted. Cover with a lid or foil until ready to use.
For the flatbreads: Heat grill to medium heat. Toss the mushrooms lightly in the olive oil and some salt and pepper. Grill the mushrooms until lightly charred, turning occasionally, about 6 minutes. Cool slightly, then remove the stems from the oyster mushrooms and slice the cremini mushrooms.
For assembly: Divide the pizza dough into two equal portions and roll them out on a lightly floured surface to 10 inches long and 4 inches wide. Lightly oil the grates of the grill. Grill the flatbreads for 2 minutes per side. Remove and add 1 cup of the truffled cheese sauce to each flatbread along with the mushrooms and Parmesan. Return the flatbreads to the grill in a low-heated area. Close the lid and cook for about 4 minutes. Sprinkle with the chives, cut each into 8 pieces and serve.