Special equipment: Oven-proof splatter screen
Preheat oven to 500 degrees F.
In a medium saute pan over medium-high heat, heat olive oil and add mushrooms and garlic. Season mushrooms with kosher salt and cook until mushrooms release their juices and begin to brown, about 6 to 8 minutes.
Lightly dust work surface with flour and roll dough into a circle, about 10 inches in diameter and about 1/2-inch thick.
Place dough on an oven-proof splatter screen and gently spread with a thin layer of tomato sauce, leaving a 1/2-inch border along the edge. Top the dough with mozzarella, mushroom mixture, and sopressata. Sprinkle the pizza with red pepper flakes. Bake until cheese has melted and the bottom of the crust is brown and crispy, about 12 to 15 minutes. Serve immediately.
In a small mixing bowl, whisk together yeast, warm water, sugar and olive oil. Let sit for about 3 minutes to fully dissolve and activate the yeast.
In a large mixing bowl, whisk together flour and salt.
With a rubber spatula, gradually mix the yeast mixture into the flour until just combined and dough barely holds together. Turn the dough out onto a clean counter surface dusted with flour.
Knead the dough with flour-dusted hands until the dough has become smooth and elastic, about 10 minutes. The surface should be tight and silky and bounce back slightly when pressed. Lightly grease a large mixing bowl with olive oil and place the dough in it.
Cover the bowl tightly with plastic wrap and set in a warm place to rise until doubled in size, 1 to 2 hours.
Once doubled, punch the dough down and divide it into 2 equal pieces. Knead each 1 slightly to form a uniform ball. Set aside and cover them with a kitchen towel or plastic wrap to rest for 30 minutes.
Dough can be made the day before and refrigerated overnight once it is divided. Keep covered tightly in plastic wrap or in a sealed food storage bag. Sauce can be made ahead of time and kept covered in the refrigerator for up to 3 days or frozen for up to 3 months.