For the dough: Stir together the sugar in 2 cups warm water in a medium bowl until dissolved. Sprinkle the yeast over the water, cover with a clean kitchen towel and let sit until the mixture becomes slightly foamy, about 5 minutes. If the mixture doesn't become foamy, it means the yeast is dead; throw the mixture out and start over with a new packet of yeast.
Add the flour and salt to the bowl of a stand mixer fitted with the dough hook and mix on low speed until combined. Add the yeast mixture and the olive oil and knead on low speed for 4 minutes. Raise the mixer speed to medium-low and knead until the dough starts to pull away from the sides of the bowl, about 6 minutes. Raise the mixer speed to medium and mix until the dough is smooth, pliable, and elastic, 5 minutes more. Lightly oil a large, clean bowl and transfer the dough to the bowl. Cover with a clean kitchen towel and let rest in a warm place until doubled in size, about 1 hour.
For the mushrooms: Meanwhile, melt 2 tablespoons of the butter in a large saute pan over medium-high heat. Add half of the portobello mushrooms, half of the oyster mushrooms, 3 sprigs thyme and half of the bay leaf to the pan. Cook, stirring occasionally, until the mushrooms no longer give off liquid and are golden brown, about 5 minutes. Add a pinch of salt, stir and transfer the mushrooms to a large bowl. Return the pan to the stovetop over medium-high heat and add the remaining 2 tablespoons butter. When it has melted, add the remaining mushrooms, thyme and bay leaf. Cook the second batch in the same manner as above. Add a pinch of salt, stir and transfer the mushrooms to the bowl with the first batch. Set aside.
To assemble: Preheat the oven to 450 degrees F. Line 2 rimmed baking sheets with parchment paper. If you'd like to grill your crust first, preheat a grill for cooking over medium heat.
Dust your work surface with flour and cut the dough in half. Using your fingertips, press out each piece into an oval roughly 12 inches long and 4 inches wide. If grilling crust first, brush the dough with olive oil and place on the grill. Grill until cooked through and well-marked on each side, 3 to 4 minutes per side. Remove from the heat and transfer the flatbreads to the prepared pans. If not grilling, simply transfer one piece of dough to each of the prepared pans and brush with olive oil.
Divide the mushrooms evenly between the two crusts and top with the truffled Pecorino and the Parmigiano-Reggiano. Bake until the cheese has melted and the flatbreads are golden brown, 8 to 10 minutes if the crusts were grilled and 17 to 20 minutes if not (if your oven can't accommodate both flatbreads on the center rack, use the upper and lower racks and rotate and switch their positions halfway through). Garnish with chives and serve immediately.
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